ring

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### Context
#### Algebra
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[[!include higher algebra - contents]]
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#### Linear algebra
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[[!include homotopy - contents]]
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# Contents
* table of contents
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## Idea
 {#Idea}

A _ring_ (also: _number ring_) is a basic structure in [[algebra]]: a [[set]] equipped with two [[binary operation | binary operations]] called _addition_ and _multiplication_, such that the operation of addition forms an [[abelian group]] and the operation of multiplication a [[monoid]] structure which [[distributivity law|distributes]] over addition.

All the familiar number systems such as the [[integer numbers]], [[rational numbers]], [[real numbers]], [[complex]] numbers are rings under the standard operations of addition and multiplication. Except for the first in this list they are indeed [[fields]], which are rings in which also the multiplication operation has an inverse for every element except 0 (the additive neutral element).

Other basic examples of rings are the [[cyclic groups]] $\mathbb{Z}_n$ under their mod-$n$ operations inherited from the integers ([[integers modulo n]]); the [[polynomial rings]], etc.

More abstractly, a [[ring]] is a [[monoid object|monoid]] [[internalization|internal to]] [[abelian groups]] (with their [[tensor product of abelian groups]]), and this perspective helps to explain the central relevance of the concept, owing to the fundamental nature of the notion of _[[monoid objects]]_. Accordingly monoids internal to other [[abelian categories]] and more generally [[stable infinity-categories]] constitute generalizations of the notion of _ring_ that are of interest. Notably when abelian groups are generalized to their analogs in [[stable homotopy theory]], namely to [[spectra]], the corresponding internal monoids are [[E-infinity rings]], a basic structure in [[higher algebra]].

Rings form a category, [[Ring]], which contains the category of commutative rings, [[CRing]], as a subcategory.

## Definition

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###### Definition

A [[ring]] (unital and not-necessarily commutative)
is an [[abelian group]] $R$ equipped with

1. an element $1 \in R$

1. a [[bilinear map]], hence a [[group homomorphism]]

   $$
     \cdot : R \otimes R \to R
   $$

   out of the [[tensor product of abelian groups]],

such that $\cdot$ is [[associativity law|associative]] and [[unit law|unital]] with respect to 1.

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###### Remark

The fact that the product is a [[bilinear map]] is the
**[[distributivity law]]**: for all $r, r_1, r_2 \in R$ we have

$$
  r \cdot (r_1 + r_2) = r \cdot r_1 + r \cdot r_2
$$

and

$$
  (r_1 + r_2) \cdot r = r_1 \cdot r + r_2 \cdot r
  \,.
$$

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+-- {: .num_prop}
###### Proposition

A (unital, non-commutative) **ring** is (equivalently)

* a [[monoid object|monoid]] [[internalization|internal to]] [[Ab]] regarded as a [[monoidal category]] equipped with the [[tensor product of abelian groups]];
* a [[pointed object|pointed]] [[enriched category|category enriched over]] [[Ab]] with a single [[object]].
* a [[ringoid]] with a single [[object]].

A **commutative** (unital) ring is a [[commutative monoid]] object in $(Ab, \otimes)$.

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###### Remark

In usual ring theory people often talk about **[[nonunital rings]]**  as well: multiplicative [[semigroups]] with additive [[abelian group]] structure where the multiplication is distributive toward addition; these are semigroup objects in $Ab$.  As in the unital case, if the semigroup is abelian then the ring is said to be **commutative nonunital**.  Note the adjective 'nonunital' is an example of the [[red herring principle]].

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+-- {: .num_remark}
###### Remark
If one removes the assumption that the additive group is abelian but retains the remaining ring axioms, the result is still a ring. More generally, this holds for nonunital rings of which multiplicative semigroup is left/right [[weakly reductive semigroup|weakly reductive]].  The result is false for arbitrary nonunital rings: for any group $(G, +, 0)$ we could define multiplication to be the $x\cdot y := 0$, and all the axioms except additive commutativity are trivially satisfied. This occurs because $\cdot$ doesn't distinguish between elements of $G$.
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## Generalizations

It is possible to [[internalization|internalise]] the notion of ring in at least two different ways.  Either one can replace the [[Set|category of sets]] in the classical definition with another category $C$ -- see [[ring object]] -- , or one can replace [[Ab]] in the fancy definition with another category $M$.

### Internalising the sets

If $C$ is a [[cartesian monoidal category]], then any [[Lawvere theory]] may be internalised in $C$.  The theory of rings is an example, so we can speak of _ring objects_ in $C$.  Then a ring object in $Set$ is simply a ring.  (This works whether your rings are unital or nonunital, commutative or noncommutative, etc.)  However, not every notion of internal ring takes this form.

The theory of rings is a combination of a monoid (or semigroup, if nonunital) and an abelian group structure. Thus, ring objects are algebras over a composed [[operad]] (or [[monad]]) of a monoid operad and an abelian group operad, using a standard [[distributive law]] for that situation in the sense of operads (or monads), which corresponds to the usual distributive law in the classical definition of a ring.

A particular example of this is a ring in a [[topos]]. In a topos one usually alternatively defines a ring object by the standard set-theoretic definition of a ring, and interpret the formulas in the sense of topos-theoretic semantics.

Picking a ring object $R$ in a [[topos]] $\mathcal{T}$ promotes it into a [[ringed topos]].

In cartesian categories one can also define the structure of an (abelian) group object as the lifting of the correspoding [[representable presheaf]] to a presheaf into (abelian) groups. This kind of lifting of some algebraic structure in sets to algebraic structure in a cartesian category makes sense when some category of algebras creates the limits needed to define them in sets.

### Internalising the abelian groups

If $M$ is a [[monoidal category]], then we can speak of [[monoid objects]] in $M$.  However, we usually want $M$ to be somewhat like $Ab$ to think of monoid objects in $M$ as internal rings.  For example, if $M$ is the category of abelian [[group objects]] in a cartesian monoidal category $C$, then we recreate the notion of ring object in $C$ from above.  Or, if $M$ is any [[Ab-enriched category]], then it behaves enough like $Ab$ that we may consider its monoid objects as internal rings.  There are yet other examples, however: a [[ring spectrum]] is a monoid object in [[spectra]], even though these are not $Ab$-enriched.

Other examples are [[simplicial ring | simplicial rings]] (as monoids in [[simplicial abelian group | simplicial abelian groups]]) and [[dg-ring | dg-rings]], as well as the $A$-rings below.

### [[ring over a ring|Rings over a ring]] ($A$-rings)

If $K$ is a commutative ring (or especially a [[field]]), then an [[associative algebra]] over $K$ is a monoid object in $K$-[[Mod]]; this is a special case of the previous section.

If $A$ is a noncommutative ring, then a __ring over $A$__, or simply an __$A$-ring__, is a monoid object $R$ in $A$-[[Bimod]] (that is, in $_A Mod _A$).  Every $A$-ring is a ring in the usual sense, in the sense that there is an obvious [[forgetful functor]] to the usual rings. In fact the unit map $A \to R$ is a morphism of rings, and the category of $A$-rings is precisely the [[coslice category]] or under-category $A/Ring$. Thus by category-theoretic rules, one might be led to unconventionally call $A$-rings "rings *under* $A$". Unfortunately, standard name for $A$-rings is "rings *over* $A$", like conventionally calling $k$-algebras the "algebras *over* $K$".

Unlike for the $k$-algebras, the multiplication $R\times R\to R$ which is the morphism of $A$-[[bimodules]], is not (left) $A$-linear in the *second* factor, but only $A^{op}$-linear (that is, $A$-linear on the right). In other words, the axiom for $K$-algebras $k (r s) = r (k s)$ is not true, for $k\in A$, $r,s\in R$, although $k (r s) = (k r) s$ and $(r s) k = r (s k)$ do hold.

Both for a discussion for under-over and also for this difference between $K$-algebras and $A$-rings see the Café\'s [quick algebra quiz](http://golem.ph.utexas.edu/category/2008/12/a_quick_algebra_quiz.html).

A dual notion to an $A$-ring is an $A$-[[coring]].

The structure of an $A\otimes A^{op}$-ring $(R,\mu,\eta)$ is determined by
the structure of $A$ as a ring, together with the two natural homomorphisms
of rings
$s = \eta(-\otimes 1_A):A\to R$ and $t=\eta(1_A\otimes -):A^{op}\to R$ which have commuting images ($s(a)t(a')=t(a')s(a)$, for all $a,a'\in A$).

### Higher rings

By replacing in the sentence "a ring is a [[monoid]] in [[Ab]]" the [[abelian category]] [[Ab]] with a [[higher category theory|higher category]] of _symmetric monoidal_ higher groupoids, one obtains higher notion of rings, such as a [[ring groupoid]], or in the commutative case, a [[symmetric ring groupoid]].

Of particular interest is the maximal case of symmetric monoidal [[∞-groupoid | ∞-groupoids]] and, even more generally, that of [[spectrum|spectra]]. A [[monoid in an (∞,1)-category]] in the [[stable (∞,1)-category of spectra]] is an [[A-infinity-ring]] or [[associative ring spectrum]]. The commutative case is a [[commutative monoid in an (∞,1)-category]]: an [[E-infinity ring]] or [[commutative ring spectrum]].

## Examples
 {#Examples}

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###### Example

* The [[integers]] $\mathbb{Z}$ are a ring under the standard addition and multiplication operation.

* For each  $n$, this induces a ring structure on the [[cyclic group]] $\mathbb{Z}_n$, given by operations in the [[integers modulo n]].

* The [[rational numbers]] $\mathbb{Q}$, [[real numbers]] $\mathbb{R}$ and [[complex numbers]] are rings under their standard operations, in fact these are even _[[fields]]_.

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###### Example

For $R$ a ring, the [[polynomials]]

$$
  r_0 + r_1 x + r_2 x^2 + \cdots + r_n x^n
$$

(for arbitrary $n \in\mathbb{N}$) in a [[variable]] $x$ with [[coefficients]] in $R$ form another ring, the _[[polynomial ring]]_ denoted $R[x]$.
This is the [[free construction|free]] $R$-[[associative algebra]] on a single generator $x$.

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###### Example

For $R$ a ring and $n \in \mathbb{N}$, the set
$M(n,R)$
of $n \times n$-[[matrices]] with [[coefficients]] in $R$
is a ring under elementwise addition and [[matrix multiplication]].

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###### Example

For $X$ a [[topological space]], the set of [[continuous functions]]
$C(X,\mathbb{R})$ or $C(X,\mathbb{C})$ with values in the
[[real numbers]] or [[complex numbers]] is a ring under pointwise
(points in $X$) addition and multiplication.

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+-- {: .num_example}
###### Example

For $X$ a [[topological space]], the [[direct sum]] of its [[ordinary cohomology]] groups $H^\bullet(X,\mathbb{Z})$ forms a ring whose multiplication operation is the [[cup product]]. This is a [[graded object|graded ring]], graded by the cohomological degree.

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## Types of rings

* [[commutative ring]]

* [[GCD ring]]

* [[Bézout ring]]

* [[Euclidean ring]]

* [[prefield ring]]

* [[Noetherian ring]]

* [[Artinian ring]]

* [[quotient ring]]

* [[composition ring]]

* [[tensor ring]]

* [[symmetric ring]]

* [[supercommutative ring]]

* [[super ring]]

* [[exterior ring]]

* [[Clifford ring]]

* [[differential ring]]

* [[Q-algebra]]

* [[ordered ring]]

* [[pseudolattice ordered ring]]

* [[totally ordered ring]]

* [[topological ring]]

* [[normed ring]]

* [[associative unital algebra]], [[commutative algebra]]

* many more...

### Integral domains

* [[integral domain]]

* [[GCD domain]]

* [[Bézout domain]]

* [[Euclidean domain]]

* [[field]]

* [[unique factorization domain]]

* [[principal ideal domain]]

* [[ordered integral domain]]

* [[archimedean integral domain]]

## Related concepts

* [[monoid]], [[monoid object]]

* [[nonunital ring]], [[nonassociative ring]]

* [[group]], [[group object]]

* [[quadratic abelian group]]

* **ring**, [[E-∞ ring]]

  * [[subring]]

  * [[ring groupoid]], [[symmetric ring groupoid]]

  * [[ring object]]

  * [[localization of a ring]]

  * [[filtered ring]], [[associated graded ring]]

  * [[w-contractible ring]]

  * [[core of a ring]]

* [[ideal]]

* [[prime ring]]

* [[spectrum of a commutative ring]]

  * [[affine scheme]], [[affine scheme]], [[spectral topological space]]

* [[ring extension]],

  * [[infinitesimal extension]]

    [[nilradical]], [[reduced ring]],

* [[Pythagorean ring]]

* [[near-ring]]

* [[module]], [[bimodule]]

## References
 {#References}

### General
 {#ReferencesGeneral}

Lecture notes:

* Arno Fehm, _Ringe_ ([pdf](http://www.math.uni-konstanz.de/~fehm/teaching/algebra/geyer2.pdf)) (in German)

Textbook accounts:

* [[Frank W. Anderson]], [[Kent R. Fuller]], _Rings and Categories of Modules_, Graduate Texts in Mathematics, **13** Springer (1992) [[doi:10.1007/978-1-4612-4418-9](https://doi.org/10.1007/978-1-4612-4418-9)]

* [[Paul Edwin Bland]], *Rings and Their Modules*, De Gruyter (2011) [[doi:10.1515/9783110250237](https://doi.org/10.1515/9783110250237), [pdf](http://site.iugaza.edu.ps/mashker/files/%D9%85%D8%AD%D8%A7%D8%B6%D8%B1%D8%A7%D8%AA-%D8%AC%D8%A8%D8%B1-%D8%AF%D9%83%D8%AA%D9%88%D8%B1%D8%A7%D8%A9.pdf)]

{#TTFormalizations} Formalization in [[univalent foundations of mathematics]] ([[homotopy type theory]] with the [[univalence axiom]]):

in a context of plain [[Agda]]:

* [[Martín Escardó]], *[Rings](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#ring-sip)*, §3.33.13 in: *Introduction to Univalent Foundations of Mathematics with Agda* [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580),  [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html)]

in a context of [[cubical type theory|cubical]] [[Agda]]:

* [[1lab]]: *[Algebra.Ring](https://1lab.dev/Algebra.Ring.html)*

### History
 {#ReferencesHistory}

[[Richard Dedekind]] had introduced the concept today called _ring_ under the name _Ordnung_ (Ger: order, as in [taxonomic order](http://en.wikipedia.org/wiki/Order_%28biology%29)). The word _Zahlring_ (Ger: number ring/ring of numbers) for this was introduced in section 9.31 of

* [[David Hilbert]], _Die Theorie der algebraischen Zahlkörper_, Jahresbericht der  Deutschen Mathematiker-Vereinigung 4 (1879)

There, the word ring just appears with a footnote mentioning Dedekind's use of the word "Ordnung", no further motivation is given. So probably Hilbert meant to use "ring" as in "collection of things holding together", not in the sense of circles or loops (as one might guess from the rings of [[cyclic groups]] $\mathbb{Z}_n$).

The first abstract axiomatic description of rings is in

* Adolf Fraenkel, Journal für die reine und angewandte Mathematik **145** (1914)

which however contains some additional axioms not used anymore. The set of axioms in its modern form appears first in

* [[Emmy Noether]], _Ideal Theory in Rings_, Mathematische Annalen **83** (1921)

For historical accounts see

* I. Kleiner, _From numbers to rings: the early history of ring theory_, Elemente der Mathematik 53 (1998) 18-35. ([web](http://dx.doi.org/10.5169/seals-3627))

* _The development of ring theory_ ([pdf](http://www-history.mcs.st-and.ac.uk/HistTopics/Ring_theory.html))
[[!redirects rings]]

[[!redirects unital ring]]
[[!redirects unital rings]]
[[!redirects ring with unit]]
[[!redirects rings with unit]]
[[!redirects rings with units]]
[[!redirects ring with identity]]
[[!redirects rings with identity]]
[[!redirects rings with identities]]

[[!redirects associative ring]]
[[!redirects associative rings]]

[[!redirects associative unital ring]]
[[!redirects associative unital rings]]

[[!redirects noncommutative ring]]
[[!redirects noncommutative rings]]

[[!redirects non-commutative ring]]
[[!redirects non-commutative rings]]

[[!redirects commutative ring]]
[[!redirects commutative rings]]

[[!redirects commutative unital ring]]
[[!redirects commutative unital rings]]

[[!redirects ring unit]]
[[!redirects ring units]]


ring extension

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## Idea

The analog of [[group extensions]] for [[rings]], classified by an analog of [[Schreier theory]].

(...)

## Properties

### Everett's theorem
 {#EverettTheorem}

(...)

## Related concepts

* [[infinitesimal extension]]

* [[Frobenius extension]]

## References

A classical textbook account is in paragraphs 52-54 of

* Rédei, _Algebra_ (1954) ([web](http://www-history.mcs.st-and.ac.uk/Extras/Redei_Algebra.html)).

A more modern account is in

* M. Petrich, _Ideal extensions of rings_, Acta Mathematica Hungaria, volume 45, numbers 3-4 (1985)
 {#Petrich}

See also

* Zachary Mesyan, _The ideals of an ideal extension_ ([arXiv:0909.0440](http://arxiv.org/abs/0909.0440))

[[!redirects ring extensions]]


ring groupoid

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#### Type theory
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# Contents
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## Idea

A *ring groupoid* is a [[1-type|1-truncated]] [[A-infinity-ring|$A_\infty$-ring]].

Equivalently: A [[groupoid]] $G$ that is both a [[symmetric 2-group]] $(G, 0, \oplus)$ and a [[monoidal groupoid]] $(G, 1, \otimes)$ such that $\otimes$ [[distributivity|distributes]] over $\oplus$ satisfying certain higher [[coherence laws]] (given in [Kelly74](#Kelly74)).

## Properties

* The [[discrete groupoid]] of [[integers]] $\mathbb{Z}$ is the [[initial object|initial]] ring groupoid.

* The [[contractible]] groupoid or the [[trivial ring groupoid]] $0$ is the [[terminal object|terminal]] ring groupoid.

## See also

* [[ring]]

* [[A-infinity-ring]]

* [[symmetric 2-group]]

* [[monoidal groupoid]]

* [[symmetric ring groupoid]]

* [[differential ring groupoid]]

* [[ring functor]], [[algebra groupoid]]

## References

* {#Kelly74} [[G. M. Kelly]], _Coherence theorems for lax algebras and distributive laws_, Lecture Notes in Mathematics **420**, Springer Verlag, Berlin(1974) 281-375 $[$[doi:10.1007/BFb0063106](https://doi.org/10.1007/BFb0063106)$]$

[[!redirects ring groupoids]]


ring object

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# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

For $C$ a [[cartesian monoidal category]] (a category with [[finite products]]), an *internal ring* or a *ring object* in $C$ is an [[internalization]] to the category $C$ of the notion of a [[ring]].

Under some reasonable assumptions on $C$ that allow one to construct a ([[symmetric monoidal category|symmetric]]) [[monoidal category|monoidal]] [[tensor product]] on the [[category]] of [[abelian group|abelian]] [[group objects]] $Ab(C)$ [[internalization|internal]] to $C$, a ring object can also be defined as a [[monoid object]] internal to that monoidal category $Ab(C)$.

Sometimes one might take this last point of view a little further, especially in certain contexts of [[stable homotopy theory]] where a [[stable (∞,1)-category]] of [[spectra]] is already something like an [[(∞,1)-category]]-analogue of a [[category of abelian groups]]. With the understanding that a [[symmetric smash product of spectra]] plays a role analogous to [[tensor products of abelian groups]], monoids with respect to the [[smash product]] are often referred to as "$xyz$-rings" of one sort or another (as mentioned at "[[ring operad]]"). Thus we have carry-over phrases from the early days of stable homotopy theory, such as "[[A-∞ rings]]" (for monoids) and "[[E-∞ rings]]" (commutative monoids). Here it is understood that the monoid multiplication on spectra is an $(\infty, 1)$-refinement of a multiplicative structure on a corresponding [[cohomology theory]], with various forms of [[K-theory]] providing archetypal examples.

## Definition
 {#Definition}

### Traditional definition

The traditional definition, based on a traditional presentation of the [[equational theory]] of rings, is that a ring object consists of an object $R$ in a [[cartesian monoidal category]] $C$ together with morphisms $a: R \times R \to R$ (addition), $m: R \times R \to R$ (multiplication), $0: 1 \to R$ (zero), $e: 1 \to R$ (multiplicative identity), $-: R \to R$ (additive inversion), subject to [[commutative diagrams]] in $C$ that express the usual ring axioms.

### As a model of a Lawvere theory
 {#AsAModelOfALawvereTheory}

Let $T$ be the [[Lawvere theory]] for rings, viz. the category [[opposite category|opposite]] to the category of [[finitely generated object|finitely generated]] [[free object|free]] rings (which are non-commutative [[polynomial rings]] $\mathbb{Z}\langle X_1, \ldots, X_n\rangle$) and ring maps between them. Then for $C$ a category with finite products, a _ring object_ in $C$ may be identified with a product-preserving functor $T \to C$.

### Via the microcosm principle
 {#ViaTheMicrocosmPrinciple}

Alternatively, one may define ring objects following the Baez–Dolan [[microcosm principle]]. Indeed, similarly to how it is possible to define monoids in a monoidal category (a pseudomonoid in $(\mathsf{Cat},\times,\mathsf{pt})$), it is possible to speak of rig objects internal to any [[bimonoidal category]] (a pseudomonoid in $(\mathsf{SymMonCats},\otimes_{\mathbb{F}},\mathbb{F})$).

Namely, a **rig in a [[bimonoidal category]]** $(\mathcal{C},\otimes_{\mathcal{C}},\oplus_{\mathcal{C}},\mathbf{0}_{\mathcal{C}},\mathbf{1}_{\mathcal{C}})$ is given by a quintuple $(R,\mu^{+}_{R},\eta^{+}_{R},\mu^{\times}_{R},\eta^{\times}_{R})$ consisting of

  - An object $R$ of $\mathcal{C}$, called the **underlying object** of the rig;
  - A morphism
    $$\mu^{+}_{R}\colon R\oplus_{\mathcal{C}}R\longrightarrow R$$
    of $\mathcal{C}$, called the **addition morphism of $R$**;
  - A morphism
    $$\mu^{\times}_{R}\colon R\otimes_{\mathcal{C}}R\longrightarrow R$$
    of $\mathcal{C}$, called the **multiplication morphism of $R$**;
  - A morphism
    $$\eta^{+}_{R}\colon\mathbf{0}_{\mathcal{C}}\longrightarrow R$$
    of $\mathcal{C}$, called the **additive unit morphism of $R$**;
  - A morphism
    $$\eta^{\times}_{R}\colon\mathbf{1}_{\mathcal{C}}\longrightarrow R$$
    of $\mathcal{C}$, called the **multiplicative unit morphism of $R$**;

satisfying the following conditions:

  1. The triple $(R,\mu^{+}_R,\eta^{+}_R)$ is a [[commutative monoid in a symmetric monoidal category | commutative monoid in $\mathcal{C}$]];
  2. The triple $(R,\mu^{\times}_R,\eta^{\times}_R)$ is a [[monoid in a monoidal category | monoid in $\mathcal{C}$]];
  3. The diagrams

     \begin{imagefromfile}
         "file_name": "semiring_in_bimonoidal_bilinearity_1.png",
         "width": 800
     \end{imagefromfile}

     corresponding to the rig axioms $a(b+c)=a b+a c$ and $(a+b)c=a c+b c$ commute;
  4. The diagrams

     \begin{imagefromfile}
         "file_name": "semiring_in_bimonoidal_bilinearity_2_corr.png",
         "width": 500
     \end{imagefromfile}

     corresponding to the rig axioms $0a=0$ and $a0=0$ commute;

Moreover, for $\mathcal{C}$ a _braided_ bimonoidal category, one defines a **commutative rig in $\mathcal{C}$** to be a rig in $\mathcal{C}$ whose multiplicative monoid structure is commutative.

A partial version of this definition first appeared in ([Brun 2006, Definition 5.1](#Brun2006)).

## Examples

* A ring object in [[Top]] is a [[topological ring]].

* A ring object in [[Ho(Top)]] is an [[H-ring]].

* A [[topos]] equipped with a ring object is called a [[ringed topos]], see there for more details.

* The [[affine line]] (see there) is a ring object in the given ambient [[topos]].

For the notion of a rig in a [[bimonoidal category]] [defined via the microcosm principle](#ViaTheMicrocosmPrinciple), we have the following examples.

 * A rig in $\left(\mathsf{Sets},\coprod,\times,\emptyset,\times\right)$ is a monoid.
 * A rig in $\left(\mathsf{CMon},\oplus,\otimes_\mathbb{N},0,\mathbb{N}\right)$ is a rig.
 * A rig in $\left(\mathsf{Ab},  \oplus,\otimes_\mathbb{Z},0,\mathbb{Z}\right)$ is a ring.
 * A rig in $\left(\mathsf{Mod}_R,\oplus,\otimes_R,0,R\right)$ is an associative algebra.
 * A rig in $\left(\mathsf{Cats},\coprod,\times,\emptyset_{\mathsf{cat}},\mathsf{pt}\right)$ is a strict monoidal category.

## Related concepts

* [[monoid]], [[monoid object]],

* [[group]], [[group object]], [[group object in an (∞,1)-category]]

* [[ring]], **ring object**

## References

* {#Brun2006} [[Morten Brun]], *Witt Vectors and Equivariant Ring Spectra*, 2006. Proceedings of the London Mathematical Society, Volume 94, pp. 351--385. ([arXiv:math/0411567](https://arxiv.org/abs/math/0411567), [doi:10.1112/plms/pdl010](https://doi.org/10.1112/plms/pdl010).)

[[!redirects ring object]]
[[!redirects ring objects]]
[[!redirects internal ring]]
[[!redirects internal rings]]

[[!redirects rig object]]
[[!redirects rig objects]]
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ring of adeles

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## Idea

The _ring of adeles_ $\mathbb{A}_k$ of any [[global field]] $k$ -- in particular of the [[rational numbers]] $\mathbb{Q}$ -- is the [[restricted product]] of all [[formal completions]] $k_v$ of $k$ at all its [[places]] $v$, where the restriction is such that only a [[finite number]] of components have [[norm]] greater than 1. (This has a useful geometric interpretation and motivation by the [[function field analogy]], more on which [below](#FunctionFieldAnalogy)).

In particular the ring of adeles of the rational numbers is equivalently the [[rationalization]] of the [[product]] of all [[p-adic integers]] (including the "prime at infinity").

In classical [[algebraic number theory]] one embeds a [[number field]] into the [[cartesian product]] of its [[completion of a ring|completions]] at its [[archimedean absolute values]]. This embedding is very useful in the [[proofs]] of several fundamental [[theorems]]. For example, the [[algebraic integers]] in the number field embed discretely and co-compactly into this [[cartesian product]], i.e., as a [[lattice in a vector space|lattice]], and this opens the way for example to the concrete realization of the [[group of units]] (modulo [[torsion subgroup|torsion]]) as a [[lattice]], and also to the technique of [[Fourier analysis]] where Poisson summation applied to the lattice has classical implications for [[theta functions]] and [[zeta functions]].

It was noticed by [[Claude Chevalley]] and [[André Weil]] that the situation is made even better if the [[number field]] itself is embedded in the cartesian product of its [[formal completions]] at all of its [[absolute values]]. With a few additional restrictions, these objects are known as the _adeles_, and the [[group of units|units]] of this ring are called the _[[ideles]]_. Under these restrictions, the given number field embeds discretely and cocompactly into the adeles, i.e., behaves as a lattice where it is again possible to apply Poisson summation.

When considering the adeles and ideles, it is their [[topology]] as much as their algebraic structure that is of interest. Many important results in [[number theory]] translate into simple statements about the topologies of the adeles and ideles. For example, the finiteness of the [[ideal class group]] and the [[Dirichlet unit theorem]] are equivalent to a certain quotient of the ideles being compact and discrete.

([Weston, p. 1](#Weston))

## Definition

The concept of a _ring of adeles_ (older synonym: ring of valuation vectors) makes sense for any [[global field]], hence for any finite-dimensional [[field extension]] of either the [[rational numbers]] or of a [[function field]] over a [[finite field]]. The ring of adeles of the [[rational numbers]] themselves is the classical case which we discuss first in

* [For the rational numbers](#ForTheRationalNumbers)

Then we consider the definition more generally

* [For a number field](#ForAnyNumberField)

and finally in full generality

* [For a global field]().

### For the rational numbers
 {#ForTheRationalNumbers}

We start off very simply with the algebraic description of the adele ring over the [[rational numbers]].

+-- {: .num_defn #RingOfAdeles}
###### Definition

The ring of _integral adeles_ $\mathbb{A}_{\mathbb{Z}}$ is the [[product]]
of the [[profinite completion of the integers]] $\widehat{\mathbb{Z}}$, with the [[real numbers]]

$$
  \mathbb{A}_{\mathbb{Z}}
  \coloneqq
  \mathbb{R} \times \widehat{\mathbb{Z}}
  \,.
$$

The _ring of adeles_ $\mathbb{A}_{\mathbb{Q}}$ (or just $\mathbb{A}$, for short) itself is the [[rationalization]] of the ring of integral adeles, hence its [[tensor product]] with the [[rational numbers]]

$$
  \mathbb{A}_\mathbb{Q}
    \coloneqq
  \mathbb{Q}
    \otimes_\mathbb{Z}
  \mathbb{A}_\mathbb{Z}
  \,.
$$

=--

This definition has various equivalent reformulations which are often useful.

+-- {: .num_remark #ProfiniteIntegersAsProductOverpAdics}
###### Remark

By [this proposition](profinite+completion+of+the+integers#AsProductOverAlsoPAdicIntegers) we have that the [[profinite completion of the integers]] is equivalently the [[product]] of all [[p-adic integers]] as $p$ ranges over all [[prime numbers]]

$$
  \hat \mathbb{Z} \simeq \underset{p\;prime}{\prod} \mathbb{Z}_p
$$

Using this in def. \ref{RingOfAdeles} says that the ring of integral adeles is the product

$$
  \mathbb{A}_{\mathbb{Z}}
    \simeq
   \mathbb{R} \times
  \underset{p\;prime}{\prod} \mathbb{Z}_p
  \,.
$$

=--

From this one obtains the following equivalent characterization:

+-- {: .num_prop #RationalRingOfAdelesAsRestrictedProduct}
###### Proposition

The ring of adeles $\mathbb{A}$, def. \ref{RingOfAdeles}, is equivalently the [[restricted product]] $\prod^\prime$ of the [[p-adic rational numbers]], the restriction being along the inclusion $\mathbb{Z}_p \to \mathbb{Q}_p$:

$$
  \mathbb{A}_{\mathbb{Q}}
    =
  \mathbb{R} \times \underset{p \; prime}{\prod^\prime} \mathbb{Q}_p
$$

=--

+-- {: .proof}
###### Proof

By remark \ref{ProfiniteIntegersAsProductOverpAdics} the
[[tensor product]] to be computed is equivalently

$$
  \mathbb{A}_{\mathbb{Q}}
  \simeq
  \mathbb{Q}\otimes_{\mathbb{Z}}
   \left(
     \mathbb{R} \times
    \underset{p \; prime}{\prod} \mathbb{Z}_p
   \right)
  \,.
$$

Now notice that a [[natural number]] $n$ is a [[group of units|unit]] in $\mathbb{Z}_p$ if $p$ is not a [[prime factor]] of $n$.
Therefore for $(a_p) \in \underset{p}{\prod} \mathbb{Z}_p$ and $\frac{c}{d} \in \mathbb{Q}$, then for each of the [[finite number]] of [[prime factors]] $p$ of $d$ the tensor product
element $\frac{c}{d} \otimes_{\mathbb{Z}} a_p \in \mathbb{Q}_p$ contains a non-vanishing negative power of $p$ and is hence not in $\mathbb{Z}_p$, whereas for all $p$ that do not appear as prime factors in $d$ it is.

=--

Finally notice:

+-- {: .num_remark #AsProductOverPlaces}
###### Remark

The [[prime numbers]] correspond to the non-archimedean [[places]] of $\mathbb{Z}$, and under this identification there is one more real [[place at infinity]], "$p = \infty$", the [[completion of a ring|completion]] of $\mathbb{Q}$ at which is the real numbers $\mathbb{R}$, which one may therefore write $\mathbb{R} = \mathbb{Q}_\infty$. Using this the characterization of the ring of adeles from prop. \ref{RationalRingOfAdelesAsRestrictedProduct} is equivalently the [[restricted product]] over all real places of the [[formal completion]] of $\mathbb{Q}$ at this place

$$
  \mathbb{A}_{\mathbb{Q}}
    \simeq
  \underset{p \in Places(\mathbb{Z})}{\prod^{\prime}} \mathbb{Q}_p
  \,.
$$

Considering this [[restricted product]] not just in bare [[commutative rings]] but in [[topological rings]] yields the right structure of a topological ring on $\mathbb{A}_{\mathbb{Q}}$. This is the content of the following proposition.

=--

+-- {: .num_prop}
###### Proposition
$\mathbb{A}_\mathbb{Q}$ is a [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] commutative ring. In particular, it is complete with respect to its [[uniform space]] structure.
=--

+-- {: .proof}
###### Proof
The restricted product is a filtered colimit of a system of *open* inclusions between locally compact Hausdorff rings, and is therefore itself locally compact Hausdorff. If $x_\alpha$ is a Cauchy net, then for all sufficiently large $\alpha, \beta$ the differences $x_\alpha - x_\beta$ lie in a compact neighborhood of the identity. Holding $\beta$ fixed, the limit $\lim_\alpha x_\alpha - x_\beta$ exists by compactness; if $x$ is this limit, then $x + x_\beta$ is the limit of the Cauchy net.
=--

If one omits the factor of $\mathbb{R} = \mathbb{Q}_\infty$, then one speaks of the _ring of finite adeles_.

$$
  \mathbb{A}_{\mathbb{Q}}^f \coloneqq \underset{p \; prime}{\prod^\prime} \mathbb{Q}_p
  \,.
$$

All of this generalizes to any [[number field]] $k$.

### For a number field
 {#ForAnyNumberField}

Let

* $k$ be a [[number field]];

* $\mathcal{O} = \mathcal{O}_k$ the [[ring of integers|ring of]] [[algebraic integers]] in $k$:

* $P$ be its [[set]] of [[places]] (equivalence classes of [[absolute values]] on $\mathcal{O}$);

* $S\subset P$ be the set of [[archimedean valuation|archimedean]] places;

* $k_v$ the [[formal completion]] of $k$ at $v\in P$.

Notice that

* for $v \in S$ then $k_v$ is isomorphic to one of the [[local fields]] $\mathbb{R}$ or $\mathbb{C}$;

* for $v \notin S$ then $k_v$ is a [[local field]] with an open compact subring $\mathcal{O}_v$ consisting of elements of [[norm]] $1$ or less.

+-- {: .num_defn}
###### Definition

For $k$ a [[number field]], the ring of _integral adeles_ $\mathbb{A}_{\mathcal{O}}$ is the [[product]] of the [[profinite completion]] $\widehat{\mathcal{O}}$ with all the archimedean completions,

$$
  \mathbb{A}_\mathcal{O}
   \coloneqq
  \widehat{\mathcal{O}} \times \prod_{v \in S} k_v
  \,,
$$

and the _ring of adeles_ over $k$ is the [[tensor product]] $\mathbb{A}_k \coloneqq k \otimes_\mathcal{O} \mathbb{A}_\mathcal{O}$.

=--

It may be shown that

$$\widehat{\mathcal{O}} \cong \prod_{v \notin S} \mathcal{O}_v$$

where each non-archimedean place $v$ may be identified with a prime ideal $p$. Now, if we view $k$ as the localization of $\mathcal{O}$ obtained by inverting all nonzero elements $x \in \mathcal{O}$, then $k$ may be written as a filtered colimit of a system of inclusions of localizations:

$$k \cong colim_x \; \mathcal{O}[x^{-1}].$$

Componentwise, we may calculate

$$\array{
\mathcal{O}[x^{-1}] \otimes_\mathcal{O} \mathcal{O}_p & = & k_p & \text{if}\; x \in p \\
 & = & \mathcal{O}_p & \text{if}\; x \notin p
}$$

and so

$$\array{
\mathcal{O}[x^{-1}] \otimes_\mathcal{O} \widehat{\mathcal{O}} & \cong & \mathcal{O}[x^{-1}] \otimes_\mathcal{O} \prod_{p \in P \backslash\; S} \mathcal{O}_p \\
 & \cong & (\prod_{x \in p} k_p) \times (\prod_{x \notin p} \mathcal{O}_p)
}$$

Putting these facts together,

$$k \otimes_\mathcal{O} \widehat{\mathcal{O}} \cong colim_x\; (\prod_{x \in p} k_p) \times (\prod_{x \notin p} \mathcal{O}_p)$$

whence

$$\begin{array}{lll}
k \otimes_\mathcal{O} \mathbb{A}_\mathcal{O} & \cong & k \otimes_\mathcal{O} \left(\prod_{v \in S} k_v \times \widehat{\mathcal{O}}\right) \\
 & \cong & colim_x \; \prod_{v \in S} k_v \times (\prod_{p: x \in p} k_p) \times (\prod_{p: x \notin p} \mathcal{O}_p)
\end{array}$$

where each component of the filtered diagram is locally compact (a product of finitely many locally compact and infinitely many compact spaces) in the [[product topology]]. Taking the filtered colimit in $Top$ over the resulting diagram of open inclusions, the result is again a locally compact ring. In this way the ring of adeles $\mathbb{A}_k$ is topologized.

+-- {: .num_remark}
###### Remark
The topology on the adele ring $\mathbb{A}_k$ is strictly finer than the subspace topology inherited from its natural inclusion into $\prod_{v \in P} k_v$ with the product topology. For example, $(\prod_{v \in S} k_v) \times \prod_{p \in P \backslash\; S} \mathcal{O}_p$ is open in the ring of adeles, but not in $\prod_{v \in P} k_v$.
=--

+-- {: .num_defn }
###### Definition

The [[group of units]] of the ring of adeles $\mathbb{A}_k$ is called the group of [[ideles]], denoted $\mathbb{I}_k$.

=--

Under the subspace topology inherited from $\mathbb{A}_k$, there is no reason for inversion $(-)^{-1}: \mathbb{I}_k \to \mathbb{I}_k$ to be continuous (and in fact it isn't!), so $\mathbb{I}_k$ isn't a topological group when topologized this way. However, we can endow $\mathbb{I}_k$ with the subspace topology given by the embedding $\mathbb{I}_k \to \mathbb{A}_k \times \mathbb{A}_k: x \mapsto (x, x^{-1})$; topologized this way, we get a locally compact topological group. _This is the topology on the ideles._

Alternatively, for each finite $T \subset P$ containing the set of archimedean places $S$, we have a locally compact group

$$\mathbb{I}_{k, T} = (\prod_{v \in T} k_v^\times) \times (\prod_{v \notin T} \mathcal{O}_v^\times)$$

(noting that each unit group $\mathcal{O}_v^\times$ is compact), and the idele group can be described as the colimit over a filtered system of open inclusions

$$\mathbb{I}_k = colim_T \; \mathbb{I}_{k, T}$$

and indeed the idele topology coincides with the filtered colimit topology.

### For a global field
 {#ForAGlobalField}

Fully generally, let $k$ be a [[global field]]. Write $P$ for its [[set]] of [[places]] and $k_v$ for its [[formal completion]] at $v \in P$.

+-- {: .num_defn #AsRestrictedProduct}
###### Definition

The _ring of adeles_ of a [[global field]] $k$ is the [[restricted product]]

$$
  \mathbb{A}_k \coloneqq \underset{v \in P}{\prod} k_v
$$

where the restriction is to elements $(x_v)_{v\in P}$ of the actual [[product]] whose components have [[norm]] at most unity -- ${\vert x_v\vert} \leq 1$, except for at most a [[finite number]] of $v$.

This is topologized in the same way as discussed [above](#ForAnyNumberField).

=--

Reviews includes ([Mathew 10](#Mathew10)).

In the function field case, where $k$ is a finite extension of $\mathbb{F}_p(T)$, the analogous ring of integers $\mathcal{O}(k)$ is the [[integral closure]] in $k$ of the subring $\mathbb{F}_p[T] \hookrightarrow \mathbb{F}_p(T)$. And analogously, the ring of integer adeles $\mathbb{A}_{\mathcal{O}(k)}$ may be defined to be the product of all the completions of $\mathcal{O}(k)$ over all the places of $k$. This is a compact ring. The restricted direct product above map may then, in parallel with the number field case described above, be described as a tensor product

$$\mathbb{A}_k = k \otimes_{\mathcal{O}(k)} \mathbb{A}_{\mathcal{O}(k)}$$

where the right side is again interpreted as a colimit in the category of topological rings of a diagram consisting of compact topological rings and open inclusions between them.

## Properties

### Basic results

+-- {: .num_prop}
###### Proposition
Under the natural inclusion $i: k \to \mathbb{A}_k$, the subspace topology on $k$ is discrete, and the quotient topology on $\mathbb{A}_k/k$ is compact.
=--

As an additive topological group, there is a natural pairing on $\mathbb{A}_k$:

$$\langle -, - \rangle: \mathbb{A}_k \times \mathbb{A}_k \to \mathbb{R}/\mathbb{Z} \cong S^1.$$

If $x = (x_v)_{v \in P}$ and $y = (y_v)_{v \in P}$, then

$$\langle x, y \rangle \coloneqq \prod_v \langle x_v, y_v \rangle_v$$

where each local pairing $\langle -, - \rangle_v$ is defined to be a composite of the form

$$k_v \times k_v \stackrel{mult}{\to} k_v \stackrel{Tr}{\to} \mathbf{Q}_v \stackrel{\chi_v}{\to} \mathbb{R}/\mathbb{Z} \cong S^1,$$

noting that a place $v$ of $k$ restricts to a place on $\mathbb{Q}$.

+-- {: .num_remarks}
###### Remarks
* The trace map $Tr$ on the finite algebraic extension $k_v/\mathbf{Q}_v$ is of course defined by $Tr(x) = \sum_\sigma \sigma(x)$ where $\sigma$ ranges over all embeddings of $k_v$ into the algebraic closure of $\mathbb{Q}_v$.

* When $v$ is the archimedean place on $\mathbb{Q}$, we will take the map $\chi_v: \mathbb{R} \to \mathbb{R}/\mathbb{Z}$ to be not the quotient map, but its *additive inverse*. For non-archimedean places $v = p$, the character $\chi_p$ is the composite $\mathbf{Q}_p \to \mathbb{Z}(p^\infty) \hookrightarrow \mathbb{R}/\mathbb{Z}$ as defined [here](http://ncatlab.org/nlab/show/p-adic+number#duality).

* For each $x, y \in \mathbb{A}_k$, observe that $\langle x_v, y_v \rangle_v = 1 \in S^1$ for all but finitely many places $v$, since $x_v, y_v \in \mathcal{O}_v$ for all but finitely many places. Hence $\langle x, y\rangle$ is well-defined.
=--

+-- {: .num_prop}
###### Proposition
The additive group $\mathbb{A}_k$ is [[Pontrjagin duality|Pontrjagin self-dual]] in the sense that the map $\phi: \mathbb{A}_k \to \mathbb{A}_k^\wedge$ induced from the pairing $\langle-, -\rangle$ is an isomorphism onto the character group.
=--

Moreover, define $\pi: \mathbb{A}_k \to k^\wedge$ to be the composite

$$\mathbb{A}_k \stackrel{\phi}{\to} \mathbb{A}_k^\wedge \stackrel{i^\wedge}{\to} k^\wedge.$$

+-- {: .num_prop}
###### Proposition
The map $\pi \circ i: k \to k^\wedge$ vanishes. The map $\mathbb{A}_k/k \to k^\wedge$ induced by $\pi: \mathbb{A}_k \to k^\wedge$ is an isomorphism of topological groups, so that $\mathbb{A}_k/k$ is Pontrjagin dual to $k$.
=--

### Function field analogy
 {#FunctionFieldAnalogy}

Via the [[function field analogy]] one may understand any [[number field]] or [[function field]] $F$ as being the [[rational functions]] on an [[arithmetic curve]] $\Sigma$. Under this identification the  ring of adeles $\mathbb{A}_F$ of $F$ has the interpretation of being the ring of functions on all punctured [[formal disks]] around all points of $\Sigma$, such that only finitely many of them do not extend to the given point. ([Frenkel 05, section 3.2](#Frenkel05)). This is most manifest in terms of def. \ref{AsRestrictedProduct} above.

This means for instance that the [[general linear group]] $GL_n(\mathbb{A}_F)$ with [[coefficients]] in the ring of adeles has the interpretation as being the [[Cech cohomology|Cech cocycles]] for [[algebraic vector bundles]] of [[rank]] $n$ on an [[algebraic curve]] with respect to any [[cover]] of that curve by the complement of a finite number of points together with the [[formal disks]] around these points. (Notice that for $n = 1$ then $GL_1(\mathbb{A}_F)$ is the [[group of ideles]]). This is part of a standard construction of the [[moduli stack of bundles]] on algebraic curves, see at _[Moduli space of bundles and the Langlands correspondence](moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence)_.

[[!include function field analogy -- table]]

## Related concepts

* [[adelic string theory]]

* [[adelic integration]]

## References

* wikipedia: [adele ring](http://en.wikipedia.org/wiki/Adele_ring), [adelic algebraic group](http://en.wikipedia.org/wiki/Adelic_algebraic_group)

* D. Goldfeld, J. Hundley, chapter 1 of _Automorphic Representations and L-functions for the General Linear Group_, vol. 1, Cambridge University
Press, 2011 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/text/gl1.pdf))

* {#Mathew10} [[Akhil Mathew]], _[The adele ring](http://amathew.wordpress.com/2010/05/21/the-adele-ring/)_, 2010

* _Adeles_ [pdf](http://wiki.epfl.ch/gant/documents/lecture2-cib2011.pdf)

* Pete Clark, _Adeles and Ideles_ ([pdf](http://math.uga.edu/~pete/8410Chapter6.pdf))

* Erwin Dassen , _Adeles & Ideles_ ([pdf](http://www.math.leidenuniv.nl/~astolk/monday/notes/dassen-adeles-ideles.pdf))

* {#Weston} Tom Weston, _The idelic approach to number theory_ ([pdf](https://drive.google.com/file/d/1PK0lPR7yq-1s1aooX4ZLr_zY4s2Ej36z/view))

* [MO/96137/categorical-description-of-the-restricted-product-adeles](http://mathoverflow.net/questions/96137/categorical-description-of-the-restricted-product-adeles)

Discussion in the context of the [[function field analogy]] and the [[geometric Langlands correspondence]] is in

* {#Frenkel05} [[Edward Frenkel]], _Lectures on the Langlands Program and Conformal Field Theory_, in _Frontiers in number theory, physics, and geometry II_, Springer Berlin Heidelberg, 2007. 387-533. ([arXiv:hep-th/0512172](http://arxiv.org/abs/hep-th/0512172))

[[!redirects rings of adeles]]

[[!redirects adele]]
[[!redirects adeles]]

[[!redirects adele ring]]
[[!redirects adele rings]]

[[!redirects integral adele]]
[[!redirects integral adeles]]

[[!redirects ring of integral adeles]]
[[!redirects rings of integral adeles]]

[[!redirects ring of valuation vectors]]
[[!redirects valuation vector]]
[[!redirects valuation vectors]]

[[!redirects ring of finite adeles]]
[[!redirects rings of finite adeles]]


ring of fractions

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

In the same vein that [[commutative rings]] are to [[integral domains]] and [[GCD rings]] are to [[GCD domains]], rings of fractions are to [[field of fractions]].

## Definition ##

Let $R$ be a [[commutative ring]]. A multiplicative submonoid of $R$ is a subset $S \subseteq R$ with an injection $m:S \hookrightarrow R$, an element $1_S \in S$ and a function $(-)\cdot_S(-):S \times S \to S$, such that $m(1_S) = 1$ and for all elements $a \in S$ and $b \in S$, $s(a \cdot_S b) = s(a) \cdot s(b)$. The [[localization of a commutative ring|localization]] of $R$ away from a multiplicative submonoid $S$ is the [[initial object|initial]] commutative ring $R[S^{-1}]$ with a commutative ring homomorphism $h:R \to R[S^{-1}]$ and a function $(-)^{-1}:S \to R[S^{-1}]$, such that for all $a \in S$, $h(m(a)) \cdot a^{-1} = 1$.

The **ring of fractions** $R[\mathrm{Reg}(R)^{-1}]$ is the localization of $R$ away from the [[multiplicative subset of regular elements]] $\mathrm{Reg}(R)$. Its elements are fractions $\frac{a}{b}$ where $a\in R$ and $b\in \mathrm{Reg}(R)$ which are by the definition the equivalence classes of pairs $(a,b) \in R\times \mathrm{Reg}(R)$ and $(a,b)\sim (c,d)$ iff $a \cdot d = b \cdot c$. The addition is given by the formula
$$\frac{a}{b}+\frac{c}{d} = \frac{a \cdot d+b \cdot c}{b \cdot d}$$
and multiplication by
$$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$

For $a \in R$, $b \in \mathrm{Reg}(R)$, $c \in \mathrm{Reg}(R)$, and $d \in \mathrm{Reg}(R)$, division is given by
$$\frac{a}{b} \div \frac{c}{d} = \frac{a \cdot d}{b \cdot c}$$

## Examples

* The ring of fractions of the [[integers]] is the [[rational numbers]] $\mathbb{Q} \coloneqq \mathbb{Z}[\mathrm{Reg}(\mathbb{Z})^{-1}]$.

* The ring of fractions of a [[Heyting integral domain]] is a [[Heyting field]].

* The ring of fractions of a [[strict approximate integral domain]] is a [[local ring]].

* The ring of fractions of any [[commutative ring]] is a [[prefield ring]].

## Related concepts

* [[Grothendieck group]]

* [[localization]]

* [[prefield ring]]

* [[multiplicative submonoid of cancellative elements]]

## References

* [[Frank Quinn]], *Proof Projects for Teachers* ([pdf](https://personal.math.vt.edu/fquinn/education/pfs4teachers0.pdf))

[[!redirects rings of fractions]]

[[!redirects fraction ring]]
[[!redirects fraction rings]]

[[!redirects prefield ring of fractions]]
[[!redirects prefield rings of fractions]]

[[!redirects quotient prefield ring]]
[[!redirects quotient prefield rings]]

[[!redirects fraction prefield ring]]
[[!redirects fraction prefield rings]]


ring of integers

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Numbers like $\sqrt{3}$ and ${}^5\sqrt{3}$ look tied to the usual integers: indeed they are obtained from an integer number by doing operation of 2nd or 5th root.
More generally, we can start with the usual integer numbers and do similar algebraic operations, namely form monic polynomials with such integer coefficients and search for roots in a larger ring, obtaining
generalized integers -- called *[[algebraic integers]]* -- as solutions of a monic polynomial equation, which makes sense in an arbitrary commutative ring.

If we look for solutions of monic equations within $\mathbb{Q}$, the field of rationals, we get nothing new, just the usual [[integers]].
Thus the __integral elements in a ring__ generalize integers in
wide context and form a "ring of integers" within a larger ring.
This generalizes the standard [[integers]] inside the [[field]] of [[rational numbers]] to various situations like [[number fields]] and [[local field|local]] [[non-archimedean fields]].

## Definition

### General definition

In any [[unital ring]] $R$ one can identify the ring $\mathbb{Z}$ with the subring $\{ n 1_R | n\in\mathbb{Z}\}\subseteq R$ of all multiples $\pm 1_R, \pm (1_R+1_R), \pm(1_R+1_R+1_R),\ldots$ of the unit element $1_R$. The ring $R$ is then a left and right [[module]] over $\mathbb{Z}$ via the multiplication with the corresponding multiple of unit, that is $n.m = (n 1_R)\cdot m$.

An __[[algebraic integer]]__ in a commutative ring $R$ is any element
$r \in R$ which satisfies [[equation]] $P(r) = 0$
where $P$ is a nontrivial [[polynomial]] whose [[coefficients]] are multiplies of $1_R$ and the top degree coefficient is $1_R$ (in other words, a [[root]] of a monic polynomial in $R$ with the coefficients in $\mathbb{Z}$).

It can be checked that the set of [[algebraic integers]]
(also said to be integral elements) in $R$ (also said the _ring of integers of_ $R$) is closed with respect to addition, multiplication and taking the negative of an element, hence a subring of $R$, which is moreover containing the usual integers $\mathbb{Z} 1_R$ as unique solutions for $x\in R$ of equations $1_R \cdot x - n 1_R = 0$.

### Notation in number theory

The subring of integers in an [[algebraic number field]] $K$ (a [[finite number|finite]]-[[dimension|dimensional]] [[field extension]] of $\mathbb{Q}$), is often denoted $\mathcal{O}_K \hookrightarrow K$. In a [[local field|local]] [[non-archimedean field]] $F$, then its ring of integers is often denoted $\mathcal{O}_F$.

## Properties in a number field
 {#OfANumberField}

Alternatively, an element $\alpha \in K$ is an __[[algebraic integer]]__ if the subring $\mathbb{Z}[\alpha] \hookrightarrow K$ generated by $\alpha$ is of [[finite number|finite]] [[rank]] over $\mathbb{Z}$. This makes it plain that the [[algebraic integers]] themselves form a [[ring]]: if $\alpha, \beta \in \mathcal{O}_K$ are integral, then both $\alpha - \beta$ and $\alpha \beta$ are contained in the image of the induced map $\mathbb{Z}[\alpha] \otimes \mathbb{Z}[\beta] \to K$, which is also of finite rank.

## Properties in a local non-archimedean field
 {#OfANonArchimedeanField}

Given a [[local field|local]] [[non-archimedean field]] $F$, then its ring of integers $\mathcal{O}_F$ is the subring of elements of [[norm]] $\leq 1$.

If $F$ is the [[formal completion]] of a [[number field]] $K$, then the ring of integers of $F$ is the formal completion of the ring of integers of $K$.

## Examples

* For $K = \mathbb{Q}$ the [[rational numbers]] then $\mathcal{O}_{\mathbb{Q}} \simeq \mathbb{Z}$ is the [[commutative ring]] of ordinary [[integers]].

* For $p$ any [[prime]] and $\mathbb{Q}_p$ the [[formal completion]] of $\mathbb{Q}$ at $p$, hence the [[p-adic numbers]], then the ring of integers of $\mathbb{Q}_p$ is $\mathbb{Z}_p$, the [[p-adic integers]].

* For $K$ the [[Gaussian numbers]] then $\mathcal{O}_K$ is the ring of [[Gaussian integers]].

* The ring of integers of a [[cyclotomic field]] $\mathbb{Q}(\zeta_n)$ is $\mathbb{Z}[\zeta_n]$, called the ring of [[cyclotomic integers]].

## Properties

### General

A ring of integers is a [[Dedekind domain]].

### Function field analogy

[[!include function field analogy -- table]]

## References

Textbook account:

* {#Cassels86} [[J. W. S. Cassels]], Section 10.3 of: *Local Fields*, Cambridge University Press, 1986 (ISBN:9781139171885, [doi:10.1017/CBO9781139171885](https://doi.org/10.1017/CBO9781139171885))

Lecture notes:

* [[James Milne]], Chapter 2 of: *Algebraic number theory*, 2020 ([pdf](https://www.jmilne.org/math/CourseNotes/ANT.pdf))

See also:

* Wikipedia, _[Ring of integers](http://en.wikipedia.org/wiki/Ring_of_integers)_

The following paper shows that the subset of integers is definable in $\mathbb{Q}$ by a universal first-order formula in the language of rings.

* Jochen Koenigsmann, _Defining $\mathbb{Z}$ in $\mathbb{Q}$_, Annals of Mathematics,  **183**, issue 1 (2016) pp 73-93, doi:[10.4007/annals.2016.183.1.2](http://dx.doi.org/10.4007/annals.2016.183.1.2), arXiv:[1011.3424](https://arxiv.org/abs/1011.3424)

[[!redirects ring of integers]]
[[!redirects rings of integers]]
[[!redirects ring of algebraic integers]]
[[!redirects rings of algebraic integers]]
[[!redirects integral element in a ring]]


ring of Witt vectors

> This article is about of groups of Witt vectors and rings of Witt vectors; which are often called just Witt groups and Witt rings. However, there is also a different notion of [[Witt group]] and [[Witt ring]].

#Contents#
* table of contents
{:toc}

## Idea

Rings of _Witt vectors_ are the [[co-free functor|co-free]] [[Lambda-rings]]. Depending on whether one defines the latter via Frobenius lifts at a single [[prime number]] $p$ one speaks of _$p$-typical Witt vectors_, or of _big Witt vectors_ if all primes are considered at once.

In [[arithmetic geometry]] the impact of rings of Witt vectors $W(R)$ of a given ring $R$ is that they are like rings [[formal power series]] on $Spec(R)$, such as rings of [[p-adic numbers]]. For more on this see at _[[arithmetic jet space]]_ and at _[[Borger's absolute geometry]]_.

In components, a _Witt vector_ is an infinite sequence of elements of a given [[commutative ring]] $k$. There is a [[ring]] structure on the set $W(k)$ of Witt vectors of $k$ and $W(k)$ is therefore called *the Witt ring of $k$*. The multiplication is defined by means of [[Witt polynomials]] $w_i$ for every [[natural number]] $i$. If the [[characteristic]] of $k$ is $0$ the Witt ring of $k$ is sometimes called *universal Witt ring* to distinguish it from the case where $k$ is of prime characteristic and a similar but different construction is of interest.

A [[Witt vector|p-adic Witt vector]] is an infinite sequence of elements af a commutative ring of [[prime]] [[characteristic]] $p$. There exists a ring structure whose construction parallels that in characteristic $0$ except that only [[Witt polynomials]]  $w_{p^l}$ whose index is a power of $p$ are taken.

More abstractly, the ring of Witt vectors carries the structure of a [[Lambda-ring]] and the construction $W \colon k\mapsto W(k)$ of the Witt ring $W(k)$ on a [[commutative ring]] $k$ is [[right adjoint]] to the [[forgetful functor]] from [[Lambda-rings]] to [[commutative rings]]. Hence rings of Witt vectors are the _[[co-free functor|co-free]] [[Lambda-rings]]_.

Moreover $W(-)$ is [[representable functor|representable]] by ring of [[symmetric function|symmetric functions]], $\Lambda$.   The reason is that $\Lambda$ is the free Lambda-ring on the commutative ring $\mathbb{Z}$.   Since $\Lambda$ is a [[Hopf algebra]], $W$ is a [[group scheme]]. This is explained at [[Lambda-ring]].

The construction of Witt vectors gives a functorial way to lift a commutative ring $A$ of [[prime]] [[characteristic]] $p$ to a commutative ring $W(A)$ of [[characteristic]] 0. Since this construction is functorial, it can be applied to the [[structure sheaf]] of an [[algebraic variety]].
In interesting special cases the resulting ring $W(A)$ has even more desirable properties: If $A$ is a [[perfect field]] then $W(A)$ is a [[discrete-valuation ring|discrete valuation ring]]. This is partly due to the fact that the construction of $W(A)$ involves a ring of [[power series]] and a ring of power series over a field is always a discrete valuation ring.

There is a generalization, $W_G$, to any profinite group, $G$, due to Dress and Siebeneicher ([DS88](#DS88)), known as [[Witt-Burnside functors|Witt-Burnside functor]].

There is a generalization to [[non-commutative Witt vectors]], however these only carry a group- but no ring structure.

The [[Lubin-Tate ring]] in [[Lubin-Tate theory]] is a [[polynomial ring]] on a ring of Witt vectors and this way Witt vectors control much of [[chromatic homotopy theory]].

## Motivation

In an *expansion* of a $p$-adic number $a=\Sigma a_i p^i$ the $a^i$ are called *digits*. Usually these digits are defined to be taken elements of the set $\{0,1,\dots,p-1\}$.

Equivalently the digits can be defined to be taken from the set $T_p:=\{x|x^{p-1}=1\}\cup \{0\}$. Elements from this set are called *Teichmüller digits* or *Teichmüller representatives*.

The set $T$ is in bijection with the [[finite field]] $F_p$. The set $W(F_p)$ of (countably) infinite sequences of elements in $F_p$ hence is in bijection to the set $\mathbb{Z}_p$ of $p$-adic integers. There is a ring structure on $W(F_p)$ called *Witt ring structure* such that all ''truncated expansion polynomials'' $\Phi_n=X^{p^n}+p X^{p^{n-1}}+p^2X^{p^{n-2}}+\dots +p^n X$ called *Witt polynomials* are morphisms

$$\Phi_n:W(F_p)\to \mathbb{Z}_p$$

of groups.

## Definition
 {#Definituion}

We first give the

* [Explicit definition in components](#DefinitionInComponents)

and then discuss the

* [Universal characterization](DefinitionUniversalCharacterization)

### In components
 {#DefinitionInComponents}

#### The ring structure

+-- {: .num_defn}
###### Definition

Let $k$ be a [[commutative ring]].

If the [[characteristic]] of $k$ is $0$ then the Witt ring $W(k)$ of $k$ is defined defined by the addition

$$(a_1, a_2, \ldots, )+(b_1, b_2, \ldots) :=(\Sigma_1(a_1,b_1), \Sigma_2(a_1,a_2,b_1,b_2), \ldots)$$

and the multiplication

$$(a_1, a_2, \ldots )\cdot (b_1, b_2, \ldots ) :=(\Pi_1(a_1,b_1), \Pi_2(a_1,a_2,b_1,b_2), \ldots )$$

If $k$ is of prime characteristic $p$ we index the defining formulas by $p^1,p^2,p^3,\dots$ instead of $1,2,3,\dots$.

Here the $\Sigma_i$  are called *addition polynomials* and the $\Pi_i$  are called *multiplication polynomials*, these are described [below](#AdditionAndMultiplicationPolynomials).

=--

#### The Witt polynomials{#Witt polynomials}, the $\Sigma_i$, the $\Pi_i$, phantom components

let $x_1, x_2, \ldots $ be a collection of [[variables]]. We can define an infinite collection of [[polynomial]]s in $\mathbb{Z}[x_1, x_2, \ldots ]$ using the following formulas:

$w_1(X)=x_1$

$w_2(X)=x_1^2+2x_2$

$w_3(X)=x_1^3+3x_3$

$w_4(X)=x_1^4+2x_2^2+4x_4$

and in general $\displaystyle w_n(X)=\sum_{d|n} dx_d^{n/d}$. The value $w_n(w)$ of the $n$-th Witt polynomial in some element $w\in W(k)$ of the Witt ring of $k$ is sometimes called the *$n$-th phantom component of $w$* or *the $n$-th ghost component of $w$.

Now let $\phi(z_1, z_2)\in\mathbb{Z}[z_1, z_2]$. This just an arbitrary two variable polynomial with coefficients in $\mathbb{Z}$.

We can define new polynomials $\Phi_i(x_1, \ldots x_i, y_1, \ldots y_i)$ such that the following condition is met $\phi(w_n(x_1, \ldots ,x_n), w_n(y_1, \ldots , y_n))=w_n(\Phi_1(x_1, y_1), \ldots , \Phi_n(x_1, \ldots x_n, y_1, \ldots , y_n))$.

In short we'll notate this $\phi(w_n(X),w_n(Y))=w_n(\Phi(X,Y))$. The first thing we need to do is make sure that such polynomials exist. Now it isn't hard to check that the $x_i$ can be written as a $\mathbb{Q}$-linear combination of the $w_n$ just by some linear algebra.

$x_1=w_1$, and $x_2=\frac{1}{2}w_2+\frac{1}{2}w_1^2$, etc. so we can plug these in to get the existence of such polynomials with coefficients in $\mathbb{Q}$. It is a fairly tedious lemma to prove that the coefficients $\Phi_i$ are actually in $\mathbb{Z}$, so we won't detract from the construction right now to prove it.

#### The addition- and multiplication polynomials
 {#AdditionAndMultiplicationPolynomials}

Define yet another set of polynomials $\Sigma_i$, $\Pi_i$ and $\iota_i$ by the following properties:

$w_n(\Sigma)=w_n(X)+w_n(Y)$, $w_n(\Pi)=w_n(X)w_n(Y)$ and $w_n(\iota)=-w_n(X)$.

We now can construct $W(A)$, the ring of generalized Witt vectors over $A$. Define $W(A)$ to be the set of all infinite sequences $(a_1, a_2, \ldots)$ with entries in $A$. Then we define addition and multiplication by $(a_1, a_2, \ldots, )+(b_1, b_2, \ldots)=(\Sigma_1(a_1,b_1), \Sigma_2(a_1,a_2,b_1,b_2), \ldots)$ and $(a_1, a_2, \ldots )\cdot (b_1, b_2, \ldots )=(\Pi_1(a_1,b_1), \Pi_2(a_1,a_2,b_1,b_2), \ldots )$.

### Universal characterization
 {#DefinitionUniversalCharacterization}

+-- {: .num_theorem}
###### Theorem

The assignment

$$
  W
  \;\colon\;
  k\mapsto W(k)
$$

is a [[functor]]

$$
  W \;\colon\; CRing \longrightarrow \Lambda Ring
$$

from the [[category]] of [[commutative rings]] to that of [[Lambda-rings]].

Composed with the [[forgetful functor]]

$$
  U \;\colon\;
  \Lambda Ring \longrightarrow CRing
$$

this is the unique [[endofunctor]] $W \;\colon\; CRing \longrightarrow CRing$ such that all [[Witt polynomials]]

$$
  w_n
  :
  \begin{cases}
    W(A)\to A
   \\
    a\mapsto w_n(a)
  \end{cases}
$$

are [[homomorphisms]] of [[rings]].

=--

+-- {: .proof}
###### Proof

There is a nice trick to prove that $W(A)$ is a ring when $A$ is a $\mathbb{Q}$-algebra. Just define $\psi: W(A)\to A^\mathbb{N}$ by $(a_1, a_2, \ldots) \mapsto (w_1(a), w_2(a), \ldots)$. This is a [[bijection]] and the addition and multiplication is taken to component-wise addition and multiplication, so since this is the standard ring structure we know $W(A)$ is a ring. Also, $w(0,0,\ldots)=(0,0,\ldots)$, so $(0,0,\ldots)$ is the additive identity, $W(1,0,0,\ldots)=(1,1,1,\ldots)$ which shows $(1,0,0,\ldots)$ is the multiplicative identity, and $w(\iota_1(a), \iota_2(a), \ldots)=(-a_1, -a_2, \ldots)$, so we see $(\iota_1(a), \iota_2(a), \ldots)$ is the additive inverse.

We can actually get this idea to work for any characteristic $0$ ring by considering the embedding $A\to A\otimes\mathbb{Q}$. We have an induced injective map $W(A)\to W(A\otimes\mathbb{Q})$. The addition and multiplication is defined by polynomials over $\mathbb{Z}$, so these operations are preserved upon tensoring with $\mathbb{Q}$. We just proved above that $W(A\otimes\mathbb{Q})$ is a ring, so since $(0,0,\ldots)\mapsto (0,0,\ldots)$ and $(1,0,0,\ldots)\mapsto (1,0,0,\ldots)$ and the map preserves inverses we get that the image of the embedding $W(A)\to W(A\otimes \mathbb{Q})$ is a subring and hence $W(A)$ is a ring.

Lastly, we need to prove this for positive [[characteristic]] rings. Choose a characteristic $0$ ring that surjects onto $A$, say $B\to A$. Then since the induced map again preserves everything and $W(B)\to W(A)$ is [[surjective]], the image is a ring and hence $W(A)$ is a ring.

=--

+-- {: .num_prop}
###### Proposition

The construction of the ring of Witt vectors $W(k)$ on a given [[commutative ring]] $k$ is the [[right adjoint]] to the [[forgetful functor]] $U$ from [[Lambda-rings]] to [[commutative rings]]

$$
  (U \dashv W)
  \;\colon\;
  CRing
  \stackrel{\overset{U}{\leftarrow}}{\underset{W}{\longrightarrow}}
  \Lambda Ring
  \,.
$$

Hence rings of Witt-vectors are the _[[co-free functors|co-free]] [[Lambda-rings]]_.

=--

This statement appears in ([Hazewinkel 08, p. 87, p. 97](#Hazewinkel08)).

+-- {: .num_remark}
###### Remark

On the other hand, the [[free construction|free]] [[Lambda-ring]] (on one generator) (hence the [[left adjoint]] construction to the [[forgetful functor]]) is the ring of [[symmetric functions]].

=--

This statement appears in ([Hazewinkel 08, p. 98](#Hazewinkel08)).

## Properties

### Operations on the p-adic Witt vectors

On untruncated $p$-adic Witt vectors there are two operations, the [[Frobenius morphism]] and the [[Verschiebung morphism]] satisfying relations ([Lemma 1](#FVrelations)) being constitutive for the definition of the [[Dieudonné ring]]: In fact the Dieudonné ring is generated by two objects satisfying these relations.

Also the $n$-truncations of a Witt ring are rings since by definition the ring operations (addition and multiplication) of the first $n$ components only involve the first $n$ components. We have $W\simeq lim_n W_n$ and the projection map $W(A)\to W_n(A)$ is a [[ring homomorphism]]. We also have operations on the truncated Witt rings.

#### The shift map

For $W(k)$ as for every $k$-scheme we have the [[Verschiebung morphism]]. It  is defined to be the adjoint operation to the [[Frobenius morphism]]. For $W(k)$ the Verschiebung morphism coincides with the shift $(a_0, a_1,\dots)\mapsto (0, a_0, a_1,\dots)$

For the truncated Witt rings and the shift operation $V:W_n(k)\to W_{n+1}(k)$ the Verschiebung morphism equals the $VR=RV$ where $R$ is the restriction map.

#### The restriction map

The restriction map $R: W_{n+1}(A)\to W_n(A)$ is given by $(a_0, \ldots, a_n)\mapsto (a_0, \ldots, a_{n-1})$.

#### The Frobenius morphism

The Frobenius endomorphism $F: W_n(A)\to W_n(A)$ is given by $(a_0, \ldots , a_{n-1})\mapsto (a_0^p, \ldots, a_{n-1}^p)$. This is also a ring map, but only because of our necessary assumption that $A$ is of characteristic $p$.

Just by brute force checking on elements we see a few relations between these operations, namely that $V(x)y=V(x F(R(y)))$ and $RVF=FRV=RFV=p$ the multiplication by $p$ map.

### Duality of finite Witt groups

For a $k$-ring $R$ let $W^\prime(R)$ denote the [[ideal]] in $W(R)$ consisting of sequences $x=(x_n)_n$ of nilpotent elements in $W(R)$ such that $x_n=0$ for large $n$.

Let $E$ denote the [[Artin-Hasse exponential]]. Then we have $E(x,1)$ is a polynomial for $x\in W^\prime(R)$ and

$$E(-,1):\begin{cases}W^\prime\to \mu_k

\\
w\mapsto E(w,1)\end{cases}$$

is a morphism of [[group schemes]] to the [[multiplicative group scheme]] $\mu_k$.

+-- {: .num_prop}
###### Proposition
a) $W^\prime (R)$ is an ideal in $W(R)$.

b) $E(-,1): W^\prime\to \mu_k$ is an morphism of [[group schemes]].

c) The morphism

$$\begin{cases}
W\times W^\prime\to \mu_k
\\
(x,y)\mapsto E(xy,1)
\end{cases}$$

is bilinear and gives an isomorphism of [[group schemes]]

$$W^\prime\to D(W)$$

where $D$ denotes the [[Cartier dual]] of $W$ (maybe it is equivalently the [[Pontryagin dual]] of the underlying group of the (plain) ring $W$). That this map is a morphism of group schemes follows from the definition of the [[Cartier dual]].

d) For $x\in W(R)$ and $y\in W^\prime (R)$ we have $E(xy,1)\in R^\times$ and

$$E(V^n x y,1)=E (T^n(x F^n y),1)=E(x F^n y,1)$$
=--

+-- {: .num_defn}
###### Definition and Theorem
Let $ker(F_n^m):=ker (F^m:W_{nk}\to W_{nk})$ denote the [[kernel]] of $m$-times iterated [[Frobenius endomorphism]] of the $n$-truncated Witt ring.Let

$$\sigma_n:\begin{cases}

W_{nk}\to W_k
\\
(\alpha_0,\dots,\alpha_{n-1})\mapsto(\alpha_0,\dots,\alpha_{n-1},0,\dots)
\end{cases}$$

be the section of the restriction $R_n:W_k\to W_{nk}$. $\sigma_n$ sends $ker(F_n^m)$ in $W^\prime$. Note that $\sigma_n$ is not a morphism of groups.

We define the bilinear map

$$\lt-,-\gt:\begin{cases}
ker(F^m_n)\times ker(F^n_m)\to R^\times
\\
\lt x,y\gt\mapsto E(\sigma_n(x)\sigma_m(y),1)
\end{cases}$$
then $\lt x,y\gt$ is bilinear and gives an isomorphism

$$ker(F^m_n)\simeq D(ker(F^n_m)$$

and satisfies

$$\lt x,t y\gt=\lt f x,y\gt$$

$$\lt x,r y\gt=\lt i x,y\gt$$

where the morphisms are

$$\array{
ker(F^m_n)
&\stackrel{t}{\to}&
ker(F^m_{n+1})
\\
\downarrow^f&&\downarrow^r
\\
ker(F^{m-1}_n)
&\stackrel{i}{\hookrightarrow}&
ker(F^m_n)
}$$

where $i$ is the canonical inclusion, and $r,f,t$ are induced by $R,F,T$ where $F$ is [[Frobenius morphism|Frobenius]], $T$ is [[Witt ring|Verschiebung]] and $R:W\to W_n$ is [[Witt ring|restriction]]. $i$ and $t$ are monomorphisms, $f$ and $r$ are epimorphisms, and for $ker(F^n_m)$ we have $F=if$ and $V=rt$.
=--

References: [Pink, §25](#Pink), [Demazure, III.4](#Demazure)

## Properties of the Witt group

The group of universal (i.e. not $p$-adic) Witt vectors equals $W(k)= 1+X k [ [ X] ]$ i.e. the multiplicative group of power series in one variable $X$ with constant term $1$.

## Properties of the Witt ring

+-- {: .num_theorem}
###### Theorem
Let $k$ be a perfect field of prime characteristic $p$.

Then $W(k)$ is a discrete valuation ring with maximal ideal generated by $p$. From the above we see that $pW(k)=(0, a_0^p, a_1^p, \ldots )$. This clearly gives $W(k)/pW(k)\simeq k$.

Also, $W(k)/p^n W(k)\simeq W_n(k)$. Thus the completion of $W(k)$ with respect to the maximal ideal is just $lim W_n(k)\simeq W(k)$ which shows that $W(k)$ is a complete discrete valuation ring.
=--

## Examples

### Basic examples

* $W_{p^\infty}(\mathbb{F}_p)\simeq \mathbb{Z}_p$ the $p$-adic integers.

* $W_{p^\infty}(\mathbb{F}_{p^n})$ is the unique [[unramified extension]] of $\mathbb{Z}_p$ of degree $n$.

### Lubin-Tate ring

The [[Lubin-Tate ring]] in [[Lubin-Tate theory]]  is a [[power series]] ring over a [[Witt ring]] and this way Witt rings govern much of [[chromatic homotopy theory]].

As an Abelian group $W(A)$ is isomorphic to the group of curves in the one-dimensional [[multiplicative group|multiplicative formal group]]. In this way there is a Witt-vector-like Abelian-group-valued functor associated to every one-dimensional formal group. For special cases, such as the Lubin–Tate formal groups, this gives rise to ring-valued functors called ramified Witt vectors. ([eom](#eom))

## Related concepts

* [[Witt Cohomology]]

* [[Lambda-ring]]

* [[Hochschild-Witt complex]] and Kaledin's non-commutative Witt vectors

* [[Lubin-Tate ring]]

* [[Borger's absolute geometry]]

* [[de Rham-Witt complex]]

## References

### Original texts and classical surveys

witt vectors were introduced in

* [[Ernst Witt]], _Zyklische Körper und Algebren der Characteristik $p$ vom Grad $p^n$. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik $p^n$_, J. Reine Angew. Math. , 176 (1936) pp. 126–140, ([web](http://www.digizeitschriften.de/main/dms/img/?IDDOC=504725))

In the context of [[formal group laws]] they were used in

* [[Jean Dieudonné]], _Groupes de Lie et hyperalgèbres de Lie sur un corps de charactéristique $p \gt 0$  VII" Math. Ann. , 134 (1957) pp. 114–133

See also

* [[Michiel Hazewinkel]], _Twisted Lubin-Tate formal group laws, ramified Witt vectors and (ramified) Artin-Hasse exponentials_, Transactions of the AMS (1980)

Surveys incluce

* [[eom]], _[Witt vector](http://www.encyclopediaofmath.org/index.php/Witt_vector)_
 {#eom}

* Michel Demazure, _[[Demazure, lectures on p-divisible groups|Lectures on p-divisible groups]]_ ([web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups))
  {#Demazure}

### Modern surveys

* [[Michiel Hazewinkel]],  _[[Hazewinkel, Witt vectors|Witt vectors]]_, ([arXiv](http://arxiv.org/abs/0804.3888))
  {#Hazewinkel08}

* [[Michiel Hazewinkel]], _Formal Groups and Applications_, review in [projecteuclid](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183548600)

* Joseph Rabinoff, _The theory of Witt vectors_ ([pdf](http://math.stanford.edu/~rabinoff/misc/witt.pdf))

* [[Richard Pink]], finite group schemes, [pdf](http://www.math.ethz.ch/~pink/ftp/FGS/CompleteNotes.pdf){#Pink}

Review in the context of the [[Kummer-Artin-Schreier-Witt exact sequence]] is in

* {#MezardRomagnyTossici11} Ariane Mézard, Matthieu Romagny, Dajano Tossici, section 2 of _Sekiguchi-Suwa theory revisited_ ([arXiv:1104.2222](http://arxiv.org/abs/1104.2222))

### Further development of the theory

* [[Dmitri Kaledin]], [[non-commutative Witt vectors]]

* [[Dmitri Kaledin]], universal Witt vectors and the ''Japanese cocycle'', [pdf](http://imperium.lenin.ru/~kaledin/math/jap.pdf)

* [[Lars Hesselholt]], [[Ib Madsen]], _On the de Rham-Witt complex in [[mixed characteristic]]_, [pdf](http://www.math.uiuc.edu/K-theory/0551/paper.pdf)

* [[Lars Hesselholt]], Witt vectors of non-commutative rings and topological cyclic homology, [pdf](http://www.math.uiuc.edu/K-theory/0135/derived.pdf)

* {#DS88} A. Dress, C. Siebeneicher, _The Burnside ring of profinite groups and
the Witt vector construction_, Adv. Math., 70, (1988), 87–132.[link](https://reader.elsevier.com/reader/sd/pii/0001870888900527?token=448709A2504AA684DD56C07A8907E9344F0C761E77B09F8B840FF05D2E8B5338DCBBCAF713EEE9E24BCB91E2FAD4883D&originRegion=eu-west-1&originCreation=20221117110955)

In the context of [[Borger's absolute geometry]]:

* {#Borger08} [[James Borger]], _The basic geometry of Witt vectors, I: The affine case_ ([arXiv:0801.1691](http://arxiv.org/abs/0801.1691))

* {#Borger10} [[James Borger]], _The basic geometry of Witt vectors, II: Spaces_ ([arXiv:1006.0092](http://arxiv.org/abs/1006.0092))

[[!redirects rings of Witt vectors]]
[[!redirects rings of Witt-vectors]]

[[!redirects Witt ring]]
[[!redirects Witt rings]]
[[!redirects Witt polynomial]]
[[!redirects Witt polynomials]]
[[!redirects group of Witt vectors]]
[[!redirects Witt vector]]
[[!redirects Witt vectors]]
[[!redirects ring of Witt vectors]]

[[!redirects big-Witt-vectors functor]]


ring operad

In mathematics, a [[highly structured ring spectrum]] or ∞-ring is an object in [[homotopy theory]] encoding a refinement of a multiplicative structure on a [[cohomology theory]]. A commutative version of an ∞-ring is called an [[E-∞ ring]]. While originally motivated by questions of geometric topology and bundle theory, they are today most often used in [[stable homotopy theory]].

For more see at

* [[A-∞ operad]]

* [[Ek-operad]]

* [[E-∞ operad]]

See also _[[ring object]]_.

[[!redirects ring operads]]


ring over a ring

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Recall that *$A$-algebras*, in the usual sense of [[associative algebras]] over a [[commutative ring]] $A$, are [[rings]] $R$ equipped with ring [[homomorphisms]] $\eta \colon A\to Z(R)$ from $A$ to their [[center]] (and a plain [[unital ring]] $R$ is an [[associative algebra]] over the [[integers]] $\mathbb{Z}$).

By the term *$A$-rings* (e.g. [Böhm (2009), §2.1](#Böhm09)) one refers to the generalization of this notion to the case where $A$ is not necessarily commutative or the [[image]] of $\eta$ is not necessarily in the center of $R$. Hence $A$-rings are general objects in the [[coslice category]] $A/Rings$ (also denoted $A \downarrow Ring$) for arbitrary $A$.

The dual notion is, in some sense, that of *$A$-corings* (e.g. [Böhm (2009), §2.2](#Böhm09)).

## Details

\begin{definition}\label{DefinitionOfARings}
For $A$ a (possibly noncommutative) [[ring]], a _ring over $A$_, or _$A$-ring_ for short, is a [[monoid object]] $R$ [[internalization|internal]] to the [[category]] [[Bimod|$A Bimod$]] (that is: ${}_A Mod_A$), understood as a [[monoidal category]] with respect to the [[tensor product of bimodules]].
\end{definition}
(e.g. [Böhm (2009), Def. 2.1](#Böhm09))

Every $A$-ring is a ring in the usual sense, in the sense that there is an obvious [[forgetful functor]] to [[Ring]]. In fact the unit map $A \to R$ is a morphism of rings, and the category of $A$-rings is precisely the [[coslice category]] or [[coslice category]] $A/Ring$. Thus by category-theoretic rules, one might be led to unconventionally call $A$-rings "rings *under* $A$". Unfortunately, standard name for $A$-rings is "rings *over* $A$", like conventionally calling $k$-algebras the "algebras *over* $K$".

Unlike for the $k$-algebras, the multiplication $R\times R\to R$ which is the morphism of $A$-[[bimodules]], is not (left) $A$-linear in the *second* factor, but only $A^{op}$-linear (that is, $A$-linear on the right). In other words, the axiom for $K$-algebras $k (r s) = r (k s)$ is not true, for $k\in A$, $r,s\in R$, although $k (r s) = (k r) s$ and $(r s) k = r (s k)$ do hold.

Both for a discussion for under-over and also for this difference between $K$-algebras and $A$-rings see the Café\'s [quick algebra quiz](http://golem.ph.utexas.edu/category/2008/12/a_quick_algebra_quiz.html).

## $A\otimes A^{op}$-rings

The structure of an $A\otimes A^{op}$-ring $(R,\mu,\eta)$ is determined by
the structure of $A$ as a ring, together with the two natural homomorphisms
of rings
$s = \eta(-\otimes 1_A):A\to R$ and $t=\eta(1_A\otimes -):A^{op}\to R$ which have commuting images ($s(a)t(a')=t(a')s(a)$, for all $a,a'\in A$). In theory of [[associative bialgebroid]]s these are the dual versions of the source and target maps from the study of [[groupoid]]s.

There is (not always associative) product of $A$-rings, and especially useful in the study of $A\otimes A^{op}$-rings, given by certain [[coend]], so called [[Takeuchi product]].

## Literature

Original articles:

* {#Takeuchi77} [[Mitsuhiro Takeuchi]], *Groups of algebras over $A \times \bar{A}$*, J. Math. Soc. Japan __29__ (1977) 459-492 [[euclid:jmsj/1240432948](http://projecteuclid.org/euclid.jmsj/1240432948), [MR0506407](https://www.ams.org/mathscinet-getitem?mr=0506407)]

Introduction and review:

* {#Böhm09} [[Gabriella Böhm]], Section 2.1 of: *Hopf algebroids*, chapter of: *Handbook of algebra* **6** (2009) 173-235 [[arxiv:math.RA/0805.3806](http://arxiv.org/abs/0805.3806), doi:10.1016/S1570-7954(08)00205-2]

See also

* [[T. Brzeziński]], G. Militaru, _Bialgebroids, $\times_{R}$-bialgebras and duality_, J. Algebra 251: 279-294, 2002, [math.QA/0012164](http://arxiv.org/abs/math/0012164)

* P. Schauenburg, _Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules_, Appl. Categ. Structures __6__ (1998), 193–222, [ps](http://www.mathematik.uni-muenchen.de/%7Eschauen/papers/bnrsthb.ps) [doi](http://dx.doi.org/10.1023/A:1008608028634)


ring spectrum

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## Idea

The terminology  _(commutative) ring spectrum_ refers either to a ([[commutative monoid|commutative]]) [[monoid]] in the [[stable homotopy category]] regarded as [[monoidal category]] via the [[smash product of spectra]], or to the richer structure of a [[monoid]] in a [[model structure for spectra]] equipped with a [[symmetric smash product of spectra]].

In the first case a ring spectrum is a [[spectrum]] equipped with a unit and product operation, which is associative, unital  (and commutative) just up to unspecified [[homotopy]], such as in an [[H-space]] structure. Accordingly, these might be called "H-ring spectra", but it is traditional to just call them "ring spectra." (An _[[H-infinity ring spectrum]]_ ([Bruner-May-McClure-Steinberger 86](#BrunerMayMcLureSteinberger86)) is such an H-ring spectrum equipped with some extra structure modeling [[extended power]] operations.)

In the second case the structure is much richer; and in good cases, such as for [[highly structured spectra]], is equivalent to [[A-∞ ring]] structure ([[E-∞ ring]] structure).

To distinguish the two situations further qualification is being used. Sometimes one says **homotopy ring spectrum** to explicitly refer to the first case (e.g. [Schwede 12, chapter II 4.1](#Schwede12)) or one says "[[highly structured ring spectrum]]" to refer explicitly to the second case. For more on this see at _[[brave new algebra]]_ and _[[higher algebra]]_.

Since the concept of [[spectrum]] is the refinement of the concept of [[abelian group]] to [[homotopy theory]]/[[(∞,1)-category theory]]. The concept of **ring spectrum** is the corresponding generalization of the notion of ([[commutative ring|commutative]]) [[ring]].

[[!include homological and higher algebra -- table]]

## Definition

For details see _[[Introduction to Stable homotopy theory]]_, _[[Introduction to Stable homotopy theory -- 1-2|Part 1-2 -- Structured spectra]]_.

## Properties

### Relation to multiplicative generalized cohomology

Under the [[Brown representability theorem]], the [[generalized cohomology theory]] represented by a ring spectrum inherits the structure of a [[multiplicative cohomology theory]].

Conversely via the [[Brown representability theorem]] a spectrum representing a [[multiplicative cohomology theory]] inherits the structure of (at least) an H-ring spectrum. See [there](multiplicative+cohomology+theory#BrownRepresentabilityByRingSpectra).

## Related concepts

* [[ultracommutative ring spectrum]]

* [[symmetric ring spectrum]]

* [[symmetric algebra spectrum]]

* [[module spectrum]], [[algebra spectrum]]

* [[Gorenstein ring spectrum]]

* [[functor with smash products]]

* [[periodic ring spectrum]]

* [[Hopf ring spectrum]]

* [[model structure for ring spectra]]

* [[Boardman homomorphism]], [[Adams spectral sequence]]

* [[multiplicative spectral sequence]]

## References

The notion of (commutative) homotopy ring spectra, i.e. ([[commutative monoid|commutative]]) [[monoids]] in the [[stable homotopy category]] with respect to the [[smash product of spectra]]:

* {#Adams74} [[Frank Adams]], part III, section 10 of _[[Stable homotopy and generalised homology]]_, 1974 ([pdf](https://www.uio.no/studier/emner/matnat/math/MAT9580/v17/documents/adams-shgh.pdf))

  > (attributed there to [[George Whitehead]])

Review:

* [[John Michael Boardman]], Sections 3,7 of: _Stable Operations in Generalized Cohomology_ [[pdf](https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf), [[Boardman-StableOperations.pdf:file]]] in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford 1995 ([doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

* {#BakerRichter04} [[Andrew Baker]], [[Birgit Richter]] _Structured ring spectra_, London Mathematical Society Lecture Notes Series 315, Springer 2004 ([ISBN:9780521603058](https://www.cambridge.org/de/academic/subjects/mathematics/algebra/structured-ring-spectra?format=PB))

* {#TamakiKono06} [[Dai Tamaki]], [[Akira Kono]], Appendix C.2 of: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([[GeneralizedCohomology.pdf:file]], [ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

* {#Malkiewich14} [[Cary Malkiewich]], section 1.3 of _The stable homotopy category_, 2014 ([pdf](http://math.stanford.edu/~carym/stable.pdf))

* [[Urs Schreiber]], _[[Introduction to Stable homotopy theory]] -- [Homotopy ring spectra](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyRingSpectra)_, 2016

* [[Birgit Richter]], _Commutative ring spectra_ ([arXiv:1710.02328](https://arxiv.org/abs/1710.02328), [pdf](https://www.math.uni-hamburg.de/home/richter/rev-commringspectra.pdf)), in: [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds.), *[[Stable categories and structured ring spectra]]*, MSRI Book Series, Cambridge University Press.

On [[H-infinity ring spectra]]:

* {#BrunerMayMcLureSteinberger86} R. Bruner, [[Peter May]], [[James McClure]], M. Steinberger, _$H_\infty$-Ring Spectra and their Applications_, Lecture Notes in Mathematics 1176, Springer 1986 ([doi:10.1007/BFb0075405](https://link.springer.com/book/10.1007/BFb0075405),  [pdf](http://www.math.uchicago.edu/~may/BOOKS/h_infty.pdf))

Discussion of connective ring spectra as monoids with respect to a smash product on [[Gamma-spaces]]:

* {#Schwede99} [[Stefan Schwede]], _Stable homotopical algebra and $\Gamma$-spaces_, Math. Proc. Camb. Phil. Soc. (1999), 126, 329 ([pdf](http://www.math.uni-bonn.de/people/schwede/Gamma.pdf))

* {#Lawson09} [[Tyler Lawson]], _Commutative Γ-rings do not model all commutative ring spectra_, Homology Homotopy Appl. Volume 11, Number 2 (2009), 189-194. ([Euclid](http://projecteuclid.org/euclid.hha/1296138517))

Discussion of ring spectra as rings with respect to the [[symmetric smash product of spectra]] on [[S-modules]] includes

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], sections 2 and 3 of _[[Modern foundations for stable homotopy theory]]_ ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

* {#EKMM} [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May|P. May]], _[[Rings, modules and algebras in stable homotopy theory]]_, AMS Mathematical Surveys and Monographs Volume 47 (1997) ([pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf))

A comprehensive account for [[symmetric spectra]] is in

* {#Schwede12} [[Stefan Schwede]], _[[Symmetric spectra]]_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

and for [[orthogonal spectra]] in

* [[Stefan Schwede]], _[[Global homotopy theory]]_

An account in terms of [[(∞,1)-category theory]] is in section 7.1 of

* [[Jacob Lurie]], _[[Higher Algebra]]_

Discussion of [[simplicial ring|simplicial]] ring spectra is in

* {#GoerssHopkins99} [[Paul Goerss]], [[Michael Hopkins]], _Simplicial Structured Ring Spectra_ (1999) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.9216))

See also the references at [[stable homotopy theory]].

* Glossary for stable and chromatic homotopy theory ([[StableChromaticGlossary.pdf:file]])

[[!redirects ring spectra]]

[[!redirects commutative ring spectrum]]
[[!redirects commutative ring spectra]]

[[!redirects homotopy ring spectrum]]
[[!redirects homotopy ring spectra]]

[[!redirects homotopy commutative ring spectrum]]
[[!redirects homotopy commutative ring spectra]]
[[!redirects homotopy-commutative ring spectrum]]
[[!redirects homotopy-commutative ring spectra]]

[[!redirects H-ring spectrum]]
[[!redirects H-ring spectra]]

[[!redirects H ring spectrum]]
[[!redirects H ring spectra]]


ring theory

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## Idea

In [[algebra]], the term '**ring theory**' usually refers to the study of classical (not necessarily commutative) [[rings]], unital or not, and often also of [[modules]] over them.

Some of the related entries in nlab include [[noetherian ring]], [[maximal spectrum]], [[prime spectrum]], [[commutative algebra]], [[prime ideal]], [[ideal]], [[filter]], [[Ore set]], [[Gabriel filter]], [[primitive ideal]], [[Schur's lemma]].

Generalizations of rings include [[rigs]], $\Omega$-[[Omega-group|groups]], [[generalized rings]] in the sense of Durov, [[associative algebra|associative algebras]] over a commutative ring; moreover [[additive categories]] can be considered as a [[horizontal categorification]] of unital rings; and once upon a time they were called _rings with many objects_.

## Related concepts

* [[diamond lemma]]


ringed site

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## Definition

A **ringed site** is a [[site]] $S_X$ equipped with a [[sheaf]] $O_X$ of [[ring]]s.

A morphism $(f^{-1}, f^\sharp):(S_X, O_X) \to (S_Y, O_Y)$ of ringed sites is a pair $(f^{-1},f^\sharp)$ where $f^{-1}:S_Y\to S_X$ is a functor representing a morphism $f:S_X\to S_Y$ of sites and $f^\sharp:O_Y\to f_* O_X$ is a morphism of sheaves of rings over $Y$ (also called a $f$-[[comorphism]]).

## Examples

* The archetypical and motivating class of examples is: $X$ a [[topological space]], $S_X = Op(X)$ the [[category of open subsets]] of $X$ with its standard [[Grothendieck topology]] and $O_X := C(-,\mathbb{R})$ the sheaf of continuous functions with values in (say) the real numbers.

* A [[supermanifold]] is a ringed site where $X$ is the underlying [[manifold]], $S_X = Op(X)$ the [[category of open subsets]] $U$ of $X$ such that for each contractible $U$ $O_X(U) \simeq C^\infty(U) \otimes_{\mathbb{R}} \Lambda^\bullet V$ for $V$ a fixed finite dimensional vector space, $\Lambda^\bullet V$ its exterior algebra and the isomorphism being one of $\mathbb{Z}_2$-grading rings.

* The full generalization of the notion of a ringed site is that of a [[structured (∞,1)-topos]].

## Related concepts

* [[ringed space]], [[locally ringed space]]

*  **ringed site**, [[locally ringed site]]

*  [[ringed topos]], [[locally ringed topos]]

* [[locally algebra-ed topos]]

* [[structured (∞,1)-topos]]

## References

* _Ringed sites_ ([web](http://stacks.math.columbia.edu/tag/04KQ)); _Locally ringed topoi_ ([web](http://stacks.math.columbia.edu/tag/04ER)); [[Aise Johan de Jong]],  _[[The Stacks Project]]_
{#deJong}

[[!redirects ringed sites]]
[[!redirects locally ringed site]]
[[!redirects locally ringed sites]]


ringed space

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## Definition

A __ringed space__ is a pair $(X,O_X)$ where $X$ is a [[topological space]] and $O_X$ is a [[sheaf]] of unital [[ring]]s. The sheaf $O_X$ is called the __[[structure sheaf]]__ of the ringed space $(X,O_X)$.

If all [[stalks]] of the structure sheaf are [[local ring]]s, it is called a [[locally ringed space]].

A __morphism of ringed spaces__ $(f,f^\sharp):(X,O_X)\to (Y,O_Y)$ is a pair
where $f:X\to Y$ is a continuous map and the __[[comorphism]]__ $f^\sharp : O_Y\to f_* O_X$ is a morphism of [[sheaf|sheaves]] of rings over $Y$. Here $f_*$ denotes the [[direct image]] [[functor]] for [[sheaf|sheaves]]. Any sheaf of abelian modules $\mathcal{M}$ equipped with actions $O_X(U)\times\mathcal{M}(U)\to\mathcal{M}(U)$ making $\mathcal{M}(U)$ left $O_X$-modules, and such that the actions strictly commute with the restrictions,
is called a __sheaf of left $O_X$-modules__.

## Remarks

* Every ringed space induces a [[ringed site]]: To a ringed space $(X,O_X)$ assign the ringed site $(Op_X,O_X)$ where $Op_X$ is the [[category of open subsets]] (with [[morphisms]] their inclusions) equipped with the [[Grothendieck pretopology]] of [[open covers]], and $O_X$ is regarded as a [[structure sheaf]] of [[rings]] on $Op_X$.

* In [toric geometry](http://en.wikipedia.org/wiki/Toric_variety) and sometimes in relation to the "absolute" [algebraic geometry over](http://en.wikipedia.org/wiki/Field_with_one_element) $F_1$, one talks about monoided or monoidal space (Kato; Deitmar); which is a topological space together with a sheaf of [[monoid]]s. N. Durov on the other hand develops a generalized algebraic geometry based on a notion of generalized ringed space, which is a space equipped with a sheaf of (commutative) generalized rings, which are finitary (= [[algebraic monad|algebraic]]) monads in $\mathrm{Set}$ with a commutativity condition (which are related to higher analogues of [[Eckmann-Hilton argument]]).

## Related concepts

* [[locally ringed space]]

* [[ringed site]], [[locally ringed site]]

* [[ringed topos]], [[locally ringed topos]]

* [[locally algebra-ed topos]]

* [[structured (∞,1)-topos]]

## References

Textbook accounts:

* [[Siegfried Bosch]], p. 247-248 in *Algebraic Geometry and Commutative Algebra*, Universitext, Springer (2017) [[doi:10.1007/978-1-4471-4829-6](https://doi.org/10.1007/978-1-4471-4829-6)]

*  Ulrich Görtz, Torsten Wedhorn, p. 55 in: *Algebraic Geometry I: Schemes*, Springer (2020) [[doi:10.1007/978-3-658-30733-2](https://doi.org/10.1007/978-3-658-30733-2)]

See also:

* {#deJong} [[Aise Johan de Jong]], _[[The Stacks Project]]_, _Ringed spaces_, ([tag 0090](https://stacks.math.columbia.edu/tag/0090))

* [[eom]], *[Ringed spaces](https://encyclopediaofmath.org/wiki/Ringed_space)*

Also see references at *[[ringed topos]]*.

[[!redirects ringed spaces]]