# Glossary of module theory

Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject.
See also: Glossary of linear algebra, Glossary of ring theory, Glossary of representation theory.
* A
* B
* C
* D
* E
* F
* G
* H
* I
* J
* K
* L
* M
* N
* P
* Q
* R
* S
* T
* U
* W
* XYZ
* See also
* References

## A
### algebraically compact

    algebraically compact module (also called pure injective module) is a module in which all systems of equations can be decided by finitary means. Alternatively, those modules which leave pure-exact sequence exact after applying Hom.

### annihilator

    1. The annihilator of a left $R$-module $M$ is the set ${\textrm {Ann}}(M):=\{r\in R~|~rm=0\,\forall m\in M\}$ . It is a (left) ideal of $R$.
    2. The annihilator of an element $m\in M$ is the set ${\textrm {Ann}}(m):=\{r\in R~|~rm=0\}$.

### Artinian

    An Artinian module is a module in which every decreasing chain of submodules becomes stationary after finitely many steps.

### associated prime

    1. associated prime

### automorphism

    An automorphism is an endomorphism that is also an isomorphism.

### Azumaya

    Azumaya's theorem says that two decompositions into modules with local endomorphism rings are equivalent.

## B
### balanced

    balanced module

### basis

    A basis of a module $M$ is a set of elements in $M$ such that every element in the module can be expressed as a finite sum of elements in the basis in a unique way.

### Beauville–Laszlo

    Beauville–Laszlo theorem

### big

    "big" usually means "not-necessarily finitely generated".

### bimodule

    bimodule

## C
### canonical module

    canonical module (the term "canonical" comes from canonical divisor)

### category

    The category of modules over a ring is the category where the objects are all the (say) left modules over the given ring and the morphisms module homomorphisms.

### character

    character module

### chain complex

    chain complex (frequently just complex)

### closed submodule

    A module is called a closed submodule if it does not contain any essential extension.

### Cohen–Macaulay

    Cohen–Macaulay module

### coherent

    A coherent module is a finitely generated module whose finitely generated submodules are finitely presented.

### cokernel

    The cokernel of a module homomorphism is the codomain quotiented by the image.

### compact

    A compact module

### completely reducible

    Synonymous to "semisimple module".

### completion

    completion of a module

### composition

    Jordan Hölder composition series

### continuous

    continuous module

### countably generated

    A countably generated module is a module that admits a generating set whose cardinality is at most countable.

### cyclic

    A module is called a cyclic module if it is generated by one element.

## D
### D

    A D-module is a module over a ring of differential operators.

### decomposition

    A decomposition of a module is a way to express a module as a direct sum of submodules.

### dense

    dense submodule

### determinant

    The determinant of a finite free module over a commutative ring is the r-th exterior power of the module when r is the rank of the module.

### differential

    A differential graded module or dg-module is a graded module with a differential.

### direct sum

    A direct sum of modules is a module that is the direct sum of the underlying abelian group together with component-wise scalar multiplication.

### dual module

    The dual module of a module M over a commutative ring R is the module $\operatorname {Hom} _{R}(M,R)$.

### dualizing

    dualizing module

### Drinfeld

    A Drinfeld module is a module over a ring of functions on algebraic curve with coefficients from a finite field.

## E
### Eilenberg–Mazur

    Eilenberg–Mazur swindle

### elementary

    elementary divisor

### endomorphism

    1. An endomorphism is a module homomorphism from a module to itself.
    2. The endomorphism ring is the set of all module homomorphisms with addition as addition of functions and multiplication composition of functions.

### enough

    enough injectives
    enough projectives

### essential

    Given a module M, an essential submodule N of M is a submodule that every nonzero submodule of M intersects non-trivially.

### exact

    exact sequence

### Ext functor

    Ext functor

### extension

    Extension of scalars uses a ring homomorphism from R to S to convert R-modules to S-modules.

## F
### faithful

    A faithful module $M$ is one where the action of each nonzero $r\in R$ on $M$ is nontrivial (i.e. $rx\neq 0$ for some $x$ in $M$). Equivalently, ${\textrm {Ann}}(M)$ is the zero ideal.

### finite

    The term "finite module" is another name for a finitely generated module.

### finite length

    A module of finite length is a module that admits a (finite) composition series.

### finite presentation

    1. A finite free presentation of a module M is an exact sequence $F_{1}\to F_{0}\to M$ where $F_{i}$ are finitely generated free modules.
    2. A finitely presented module is a module that admits a finite free presentation.

### finitely generated

    A module $M$ is finitely generated if there exist finitely many elements $x_{1},...,x_{n}$ in $M$ such that every element of $M$ is a finite linear combination of those elements with coefficients from the scalar ring $R$.

### fitting

    1. fitting ideal
    2. Fitting's lemma

### five

    Five lemma

### flat

    A $R$-module $F$ is called a flat module if the tensor product functor $-\otimes _{R}F$ is exact. In particular, every projective module is flat.

### free

    A free module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the scalar ring $R$.

### Frobenius reciprocity

    Frobenius reciprocity.

## G
### Galois

    A Galois module is a module over the group ring of a Galois group.

### generating set

    A subset of a module is called a generating set of the module if the submodule generated by the set (i.e., the smallest subset containing the set) is the entire module itself.

### global

    global dimension

### graded

    A module $M$ over a graded ring $A=\bigoplus _{n\in \mathbb {N} }A_{n}$ is a graded module if $M$ can be expressed as a direct sum $\bigoplus _{i\in \mathbb {N} }M_{i}$ and $A_{i}M_{j}\subseteq M_{i+j}$.

## H
### Herbrand quotient

    A Herbrand quotient of a module homomorphism is another term for index.

### Hilbert

    1. Hilbert's syzygy theorem
    2. The Hilbert–Poincaré series of a graded module.
    3. The Hilbert–Serre theorem tells when a Hilbert–Poincaré series is a rational function.

### homological dimension

    homological dimension

### homomorphism

    For two left $R$-modules $M_{1},M_{2}$, a group homomorphism $\phi :M_{1}\to M_{2}$ is called homomorphism of $R$-modules if $r\phi (m)=\phi (rm)\,\forall r\in R,m\in M_{1}$ .

### Hom

    Hom functor

## I
### idempotent

    An idempotent is an endomorphism whose square is itself.

### indecomposable

    An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules. Every simple module is indecomposable (but not conversely).

### index

    The index of an endomorphism $f:M\to M$ is the difference $\operatorname {length} (\operatorname {coker} (f))-\operatorname {length} (\operatorname {ker} (f))$, when the cokernel and kernel of $f$ have finite length.

### injective

    1. A $R$-module $Q$ is called an injective module if given a $R$-module homomorphism $g:X\to Q$, and an injective $R$-module homomorphism $f:X\to Y$, there exists a $R$-module homomorphism $h:Y\to Q$ such that $f\circ h=g$. The module Q is injective if the diagram commutes
       The following conditions are equivalent:
      * The contravariant functor ${\textrm {Hom}}_{R}(-,I)$ is exact.
      * $I$ is a injective module.
      * Every short exact sequence $0\to I\to L\to L'\to 0$ is split.
    2. An injective envelope (also called injective hull) is a maximal essential extension, or a minimal embedding in an injective module.
    3. An injective cogenerator is an injective module such that every module has a nonzero homomorphism into it.

### invariant

    invariants

### invertible

    An invertible module over a commutative ring is a rank-one finite projective module.

### irreducible module

    Another name for a simple module.

### isomorphism

    An isomorphism between modules is an invertible module homomorphism.

## J
### Jacobson

    Jacobson's density theorem

## K
### Kähler differentials

    Kähler differentials

### Kaplansky

    Kaplansky's theorem on a projective module says that a projective module over a local ring is free.

### kernel

    The kernel of a module homomorphism is the pre-image of the zero element.

### Koszul complex

    Koszul complex

### Krull–Schmidt

    The Krull–Schmidt theorem says that (1) a finite-length module admits an indecomposable decomposition and (2) any two indecomposable decompositions of it are equivalent.

## L
### length

    The length of a module is the common length of any composition series of the module; the length is infinite if there is no composition series. Over a field, the length is more commonly known as the dimension.

### linear

    1. A linear map is another term for a module homomorphism.
    2. Linear topology

### localization

    Localization of a module converts R modules to S modules, where S is a localization of R.

## M
### Matlis module

    Matlis module

### Mitchell's embedding theorem

    Mitchell's embedding theorem

### Mittag-Leffler

    Mittag-Leffler condition (ML)

### module

    1. A left module $M$ over the ring $R$ is an abelian group $(M,+)$ with an operation $R\times M\to M$ (called scalar multipliction) satisfies the following condition:
          $\forall r,s\in R,\forall m,n\in M$,
      * $r(m+n)=rm+rn$
      * $r(sm)=(rs)m$
      * $1_{R}\,m=m$
    2. A right module $M$ over the ring $R$ is an abelian group $(M,+)$ with an operation $M\times R\to M$ satisfies the following condition:
          $\forall r,s\in R,\forall m,n\in M$,
      * $(m+n)r=mr+nr$
      * $(ms)r=r(sm)$
      * $m1_{R}=m$
    3. All the modules together with all the module homomorphisms between them form the category of modules.

### module spectrum

    A module spectrum is a spectrum with an action of a ring spectrum.

## N
### nilpotent

    A nilpotent endomorphism is an endomorphism, some power of which is zero.

### Noetherian

    A Noetherian module is a module such that every submodule is finitely generated. Equivalently, every increasing chain of submodules becomes stationary after finitely many steps.

### normal

    normal forms for matrices

## P
### perfect

    1. perfect complex
    2. perfect module

### principal

    A principal indecomposable module is a cyclic indecomposable projective module.

### primary

    primary submodule

### projective

    The characteristic property of projective modules is called lifting.A $R$-module $P$ is called a projective module if given a $R$-module homomorphism $g:P\to M$, and a surjective $R$-module homomorphism $f:N\to M$, there exists a $R$-module homomorphism $h:P\to N$ such that $f\circ h=g$.
       The following conditions are equivalent:
      * The covariant functor ${\textrm {Hom}}_{R}(P,-)$ is exact.
      * $M$ is a projective module.
      * Every short exact sequence $0\to L\to L'\to P\to 0$ is split.
      * $M$ is a direct summand of free modules.
       In particular, every free module is projective.
    2. The projective dimension of a module is the minimal length of (if any) a finite projective resolution of the module; the dimension is infinite if there is no finite projective resolution.
    3. A projective cover is a minimal surjection from a projective module.

### pure submodule

    pure submodule

## Q
### Quillen–Suslin theorem

    The Quillen–Suslin theorem states that a finite projective module over a polynomial ring is free.

### quotient

    Given a left $R$-module $M$ and a submodule $N$, the quotient group $M/N$ can be made to be a left $R$-module by $r(m+N)=rm+N$ for $r\in R,\,m\in M$. It is called a quotient module or factor module.

## R
### radical

    The radical of a module is the intersection of the maximal submodules. For Artinian modules, the smallest submodule with semisimple quotient.

### rational

    rational canonical form

### reflexive

    A reflexive module is a module that is isomorphic via the natural map to its second dual.

### resolution

    resolution

### restriction

    Restriction of scalars uses a ring homomorphism from R to S to convert S-modules to R-modules.

## S
### Schanuel

    Schanuel's lemma

### Schur

    Schur's lemma says that the endomorphism ring of a simple module is a division ring.

### Shapiro

    Shapiro's lemma

### sheaf of modules

    sheaf of modules

### snake

    snake lemma

### socle

    The socle is the largest semisimple submodule.

### semisimple

    A semisimple module is a direct sum of simple modules.

### simple

    A simple module is a nonzero module whose only submodules are zero and itself.

### Smith

    Smith normal form

### stably free

    A stably free module

### structure theorem

    The structure theorem for finitely generated modules over a principal ideal domain says that a finitely generated modules over PIDs are finite direct sums of primary cyclic modules.

### submodule

    Given a $R$-module $M$, an additive subgroup $N$ of $M$ is a submodule if $RN\subset N$.

### support

    The support of a module over a commutative ring is the set of prime ideals at which the localizations of the module are nonzero.

## T
### tensor

    Tensor product of modules

### topological

    A topological module

### Tor

    Tor functor

### torsion-free

    torsion-free module

### torsionless

    torsionless module

## U
### uniform

    A uniform module is a module in which every two non-zero submodules have a non-zero intersection.

## W
### weak

    weak dimension

## Z
### zero

    1. The zero module is a module consisting of only zero element.
    2. The zero module homomorphism is a module homomorphism that maps every element to zero.