hcju/proofwikips huggingface dataset:
Definition:Divisor (Algebra)/Terminology
Let $x \divides y$ denote that '''$x$ divides $y$'''. Then the following terminology can be used:  : $x$ is a '''divisor''' of $y$ : $y$ is a '''multiple''' of $x$ : $y$ is '''divisible by''' $x$. In the field of Euclidean geometry, in particular: : $x$ '''measures''' $y$. To indicate that $x$ does ''not'' divide $y$, we write $x \nmid y$.
Definition:Divisor (Algebra)/Gaussian Integer
Let $\struct {\Z \left[{i}\right], +, \times}$ be the ring of Gaussian integers. Let $x, y \in \Z \left[{i}\right]$. Then '''$x$ divides $y$''' is defined as: :$x \divides y \iff \exists t \in \Z \left[{i}\right]: y = t \times x$
Definition:Spectrum of Ring Functor
Let $\mathbf{Ring}$ be the category of commutative rings with unity. === To topological spaces === Let $\mathbf{Top}$ be the category of topological spaces. The '''spectrum functor''' $\Spec {} : \mathbf{Ring} \to \mathbf{Top}$ is the contravariant functor with: {{DefineFunctor |ob = $\Spec A$ is the spectrum of a ring $A$ |mor = If $f : A \to B$ is a ring homomorphism, $\Spec f : \Spec B \to \Spec A$ is the  induced map on spectra }} {{definition wanted|to schemes}}
Definition:Ring Antirepresentation
Let $R$ be a ring. Let $M$ be an abelian group. A '''ring antirepresentation''' of $R$ on $M$ is a ring antihomomorphism to the endomorphism ring of $M$.
Definition:Ring Action Defined by Ring Representation
Let $R$ be a ring. Let $M$ be an abelian group. Let $\rho : R \to \map {\operatorname {End} } M$ be a ring representation. The '''associated (left) ring action''' is the linear ring action: :$R \times M \to M$: :$\tuple {r, m} \mapsto \map {\map \rho r} m$
Definition:Ring Representation/Unital
Let $R$ be a ring with unity. Let $M$ be an abelian group. A '''unital ring representation''' of $R$ on $M$ is a ring representation $R \to \map {\operatorname {End} } M$ which is unital. That is, it is a unital ring homomorphism from $R$ to the endomorphism ring $\map {\operatorname {End} } M$.
Definition:Closed Set Axioms
Let $S$ be a set. The '''closed set axioms''' are the conditions under which a subset $F \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ consists of the closed sets of a topology on $S$: {{begin-axiom}} {{axiom | n = C1         | t = The intersection of an arbitrary subset of $F$ is an element of $F$. }} {{axiom | n = C2         | t = The union of any two elements of $F$ is an element of $F$. }} {{axiom | n = C3         | t = $\varnothing$ is an element of $F$. }} {{end-axiom}}
Definition:Topology Defined by Closed Sets
Let $S$ be a set. Let $F \subseteq \mathcal P(S)$ be a subset of its power set satisfying the closed set axioms. The '''topology defined by $F$''' is the topology whose open sets are the complements of elements of $F$: :$\tau = \{U \subseteq S : S \setminus U \in F\}$
Definition:Two-Sided Linear Combination in Ring
Let $R$ be a ring. Let $\family {x_i}_{i \mathop \in I}$ be a family of elements of $R$. A '''two-sided linear combination''' of the family is an element of the form: :$\displaystyle \sum_{i \mathop \in I} a_i x_i b_i$ where: :$\family {a_i}_{i \mathop \in I}$ and $\family {b_i}_{i \mathop \in I}$ are families in $R$ of finite support :$\sum$ denotes summation with finite support
Definition:Prime Ideal of Number Field
Let $K$ be a number field. Let $\mathcal O_K$ be its ring of integers. Let $\mathfrak p \subseteq \mathcal O_K$ be an ideal. Then $\mathfrak p$ is a '''prime ideal'''  it is not the unit ideal $(1)$ and $\mathfrak p$ has no divisors other than $\mathfrak p$ and $(1)$.
Definition:Rational Integer
Let $K$ be a number field. A '''rational integer''' of $K$ is an integer of $K$, that is, an element of $\mathbb Z \subseteq K$.
Definition:Dedekind Domain
=== Definition 1 === {{Definition:Dedekind Domain/Definition 1}} === Definition 2 === {{Definition:Dedekind Domain/Definition 2}} === Definition 3 === {{Definition:Dedekind Domain/Definition 3}} === Definition 4 === {{Definition:Dedekind Domain/Definition 4}} === Definition 5 === {{Definition:Dedekind Domain/Definition 5}} === Definition 6 === {{Definition:Dedekind Domain/Definition 6}}
Definition:Integrally Closed Integral Domain
Let $R$ be an integral domain. Then $R$ is '''integrally closed'''  it is integrally closed in its field of fractions.
Definition:Integrally Closed in Ring Extension
Let $\phi : A \hookrightarrow B$ be a ring extension. Let $C$ be the integral closure of $A$ in $B$. Then $A$ is '''integrally closed''' in $B$  $C = \phi(A)$.
Definition:Fractional Ideal
Let $R$ be an integral domain with field of fractions $K$. A '''fractional ideal''' of $R$ is a subset of $K$ that is a product of an ideal $I$ with the inverse $\lambda^{-1}$ of an element of $R$: :$\lambda^{-1} I = \left\{ {\lambda^{-1} a : a \in I}\right\}$
Definition:Dedekind Domain/Definition 1
A '''Dedekind domain''' is an integral domain in which every nonzero proper ideal has a prime ideal factorization that is unique up to permutation of the factors. {{explain|unique up to permutation}}
Definition:Dedekind Domain/Definition 2
A '''Dedekind domain''' is an integral domain of which every nonzero fractional ideal is invertible.
Definition:Dedekind Domain/Definition 3
A '''Dedekind domain''' is a noetherian domain of dimension $1$ that is integrally closed.
Definition:Dedekind Domain/Definition 4
A '''Dedekind domain''' is a noetherian domain of dimension $1$ in which every primary ideal is the power of a prime ideal.
Definition:Dedekind Domain/Definition 5
A '''Dedekind domain''' is a noetherian domain $A$ of dimension $1$ such that for every maximal ideal $\mathfrak p$, the localization $A_{\mathfrak p}$ is a discrete valuation ring.
Definition:Dedekind Domain/Definition 6
A '''Dedekind domain''' is a Krull domain of dimension $1$.
Definition:Polynomial over Ring/Multiple Variables
Let $I$ be a set. A '''polynomial over $I$ in one variable''' is an element of a polynomial ring in $I$ variables over $R$. Thus: :''Let $P \in R \left[{\left\langle{X_i}\right\rangle_{i \mathop \in I} }\right]$ be a polynomial'' is a short way of saying: : Let $R \left[{\left\langle{X_i}\right\rangle_{i \mathop \in I} }\right]$ be a polynomial ring in $I$ variables over $R$, call its family of variables $\left\langle{X_i}\right\rangle_{i \mathop \in I}$, and let $P$ be an element of this ring.
Definition:Digamma Function
The '''digamma function''', $\psi$, is defined, for $z \in \C \setminus \Z_{\le 0}$, by the logarithmic derivative of the gamma function:  :$\map \psi z = \dfrac {\map {\Gamma'} z} {\map \Gamma z}$ where $\Gamma$ is the gamma function, and $\Gamma'$ denotes its derivative.
Definition:Generated Ideal of Ring/Definition 1
The '''ideal generated by $S$''' is the smallest ideal of $R$ containing $S$, that is, the intersection of all ideals containing $S$.
Definition:Generated Ideal of Ring/Definition 2
Let $R$ be a commutative ring with unity.
Definition:Product of Fractional Ideals
Let $R$ be an integral domain with fraction field $K$. Let $I, J \subseteq K$ be fractional ideals of $R$. The '''product of $I$ and $J$''' is the set of summations: :$\left\{ \displaystyle\sum_{i = 1}^r a_i b_i : a_i \in I, b_i \in J, r \in \N \right\}$
Definition:Invertible Fractional Ideal
Let $R$ be an integral domain with fraction field $K$. Let $I\subseteq K$ be a fractional ideal of $R$. Then $I$ is '''invertible'''  there exists a fractional ideal $J\subseteq K$ such that their product is the unit ideal of $R$: :$I J = \ideal 1$
Definition:Minimal Condition/Ordered Set
Let $(P, \leq)$ be an ordered set. Then $P$ satisfies the '''minimal condition'''  it is '''well-founded''': :Every non-empty subset has a minimal element.
Definition:Coprime Ideals
Let $A$ be a commutative ring with unity. Let $I, J \subseteq A$ be ideals. Then $I$ and $J$ are '''coprime'''  their sum is the unit ideal: :$I + J = A$.
Definition:Unit Ideal
Let $A$ be a commutative ring with unity. Its '''unit ideal''' is $A$ itself.
Definition:Saturated Multiplicatively Closed Subset of Ring
Let $A$ be a commutative ring with unity. Let $S \subseteq A$ be a multiplicatively closed subset. Then $S$ is '''saturated'''  it equals its saturation, that is: :$x, y \in A, xy \in S \implies x,y \in S$
Definition:Contraction of Ideal
Let $A$ and $B$ be commutative ring with unity. Let $f : A \to B$ be a ring homomorphism. Let $\mathfrak b \subseteq B$ be an ideal. The '''contraction''' of $\mathfrak b$ by $f$ is its preimage under $f$: :$\mathfrak b^c = f^{-1}(\mathfrak b)$
Definition:Canonical Homomorphism from Ring to Unital Algebra
Let $A$ be a commutative ring with unity. Let $B$ be an unital algebra over $A$ with unit $1$ whose underlying module is unitary. The '''canonical homomorphism''' $A \to B$ is the algebra homomorphism that sends $a \in A$ to $a \cdot 1$.
Definition:Algebra Homomorphism
Let $R$ be a commutative ring. Let $(A, *)$ and $(B, \times)$ be algebras over $R$. An '''algebra homomorphism''' $f : A \to B$ is a module homomorphism such that: :$\forall a, b \in A : f(a * b) = f(a) \times f(b)$
Definition:Underlying Module of Algebra
Let $R$ be a commutative ring. Let $(A, *)$ be an algebra over $R$. Its '''underlying module''' is the $R$-module $A$.
Definition:Unital Algebra Homomorphism
Let $R$ be a commutative ring. Let $\left({A, *}\right)$ and $\left({B, \times}\right)$ be unital algebras over $R$ with units $1_A$ and $1_B$. A '''unital algebra homomorphism''' $f : A \to B$ is a algebra homomorphism such that $f \left({1_A}\right) = 1_B$.
Definition:Riemann P-symbol
The '''Riemann P-symbol''', written:  :$\displaystyle \map f z = \operatorname P \set {\begin{matrix} a & b & c \\ \alpha & \beta & \gamma & z \\ \alpha' & \beta' & \gamma' \end{matrix} }$ denotes the solutions to the hypergeometric differential equation:  {{begin-eqn}} {{eqn | o =        | r = \frac {\d^2 f} {\d z^2} }} {{eqn | o =        | ro= +       | r = \paren {\frac {1 - \alpha - \alpha'} {z - a} + \frac {1 - \beta - \beta'} {z - b} + \frac {1 - \gamma - \gamma'} {z - c} } \frac {\d f} {\d z} }} {{eqn | o =        | ro= +       | r = \paren {\frac {\alpha \alpha' \paren {a - b} \paren {a - c} } {z - a} + \frac {\beta \beta' \paren {b - c} \paren {b - a} } {z - b} + \frac {\gamma \gamma' \paren {c - a} \paren {c - b} } {z - c} } \frac f {\paren {z - a} \paren {z - b} \paren {z - c} } }} {{eqn | r = 0 }} {{end-eqn}} where: :$\alpha + \alpha' + \beta + \beta' + \gamma + \gamma' = 1$
Definition:Induced Homomorphism between Localizations of Ring
Let $A$ be a commutative ring with unity. Let $S, T \subseteq A$ be multiplicatively closed subsets. Let $S$ be a subset of the saturation of $T$. The '''induced homomorphism''' between localizations $A_S \to A_T$ is the unique $A$-algebra homomorphism between them.
Definition:Principal Open Subset of Spectrum
Let $A$ be a commutative ring with unity. Let $f \in A$. The '''principal open subset''' determined by $f$ of the spectrum $\operatorname{Spec} A$ is the complement of the vanishing set $V \left({f}\right)$: :$D \left({f}\right) = \operatorname{Spec} A - V \left({f}\right)$ That is, it is the set of prime ideals $\mathfrak p \subseteq A$ with $f \notin \mathfrak p$.
Definition:Unital Associative Commutative Algebra Homomorphism
Let $A$ be a commutative ring with unity. Let $B$ and $C$ be $A$-algebras. === Definition 1 === Let $B$ and $C$ be viewed as rings under $A$, say $(B, f)$ and $(C, g)$. An '''$A$-algebra homomorphism''' $h : B \to C$ is a morphism of rings under $A$. That is, a unital ring homomorphism $h$ such that $g = h \circ f$: :$\xymatrix{ A \ar[d]_f \ar[r]^{g} & C\\ B \ar[ru]_{h} }$ === Definition 2 === Let $B$ and $C$ be viewed as unital algebras over $A$. An '''$A$-algebra homomorphism''' $h : B \to C$ is a unital algebra homomorphism.
Definition:Radical of Ideal of Ring/Definition 1
The '''radical of $I$''' is the ideal of elements of which some power is in $I$: :$\map {\operatorname {Rad} } I := \set {a \in A: \exists n \in \N : a^n \in I}$
Definition:Radical of Ideal of Ring/Definition 2
Let $A / I$ be the quotient ring. Let $\Nil {A / I}$ be its nilradical. Let $\pi: A \to A / I$ be the quotient mapping. The '''radical''' of $I$ is the preimage of $\Nil {A / I}$ under $\pi$: :$\map {\operatorname {Rad} } I = \pi^{-1} \sqbrk {\Nil {A / I} }$
Definition:Finitely Generated Algebra
Let $A$ be a commutative ring. Let $B$ be an $A$-algebra. Then $B$ is '''finitely generated'''  $B$ has a generator which is finite.
Definition:Characteristic Polynomial of Linear Operator
Let $A$ be a commutative ring with unity. Let $M$ be a free module over $A$ of finite rank $n > 0$. Let $\phi : M \to M$ be a linear operator. === Definition 1 === The '''characteristic polynomial''' of $\phi$ is the characteristic polynomial of the relative matrix of $\phi$ with respect to a basis of $M$. === Definition 2 === Let $A[x]$ be the polynomial ring in one variable over $A$. Let $\operatorname{id}$ be the identity mapping on $M$. Let $M \otimes_A A[x]$ be the extension of scalars of $M$ to $A[x]$. The '''characteristic polynomial''' of $\phi$ is the determinant of the linear operator $x\operatorname{id} - \phi$ on $M \otimes_A A[x]$.
Definition:Induced Mapping on Maximal Spectra
Let $k$ be a field. Let $A$ and $B$ be finitely generated $k$-algebras. Let $f : A \to B$ be a $k$-algebra homomorphism. The '''induced mapping on spectra by $f$''' is the mapping $f^* : \operatorname{Max} B \to \operatorname{Max} A$ between their maximal spectra with: :$f^* (\mathfrak m) = f^{-1}(\mathfrak m)$, the preimage of a maximal ideal $\mathfrak m \in \operatorname{Max} B$. === Induced morphism of locally ringed spaces ===
Definition:Finite Algebra over Ring
Let $A$ be a commutative ring with unity. Let $B$ be an $A$-algebra. Then $B$ is '''finite''' over $A$  its underlying module is finitely generated.
Definition:Module Defined by Ring Homomorphism
Let $A$ and $B$ be a rings. Let $f: A \to B$ be a ring homomorphism. === Definition 1 === The '''left $A$-module structure''' of $B$ via $f$ is the module with left ring action: :$A \times B \to B$ :$ \tuple {a, b} \mapsto \map f a \cdot b$ === Definition 2 === The '''left $A$-module structure''' of $B$ via $f$ is the restriction of scalars of the $B$-module structure of $B$. === Definition 3 === Let $\lambda: B \to \map {\operatorname {End} } B$ be its left regular ring representation. The '''left $A$-module structure''' of $B$ via $f$ is the module with ring representation the composition $\lambda \circ f$.
Definition:Weierstrass's Elliptic Function
'''Weierstrass's Elliptic Function''' is an elliptic function, given for all complex $z$ (except for $z \in \set {2 m \omega_1 + 2 n \omega_2: \tuple {n, m} \in \Z^2}$ where the function has double poles, by Poles of Weierstrass's Elliptic Function) by:  :$\displaystyle \map \wp {z; \omega_1, \omega_2} = \frac 1 {z^2} + {\sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } } \paren {\frac 1 {\paren {z - 2 m \omega_1 - 2 n \omega_2}^2} - \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^2} }$ where $\omega_1$ and $\omega_2$ are non-zero complex constants with $\dfrac {\omega_1} {\omega_2}$ having a positive imaginary part.
Definition:Right-Hand Derivative/Real Function
Let $f: \R \to \R$ be a real function. The '''right-hand derivative''' of $f$ is defined as the right-hand limit: :$\displaystyle \map {f'_+} x = \lim_{h \mathop \to 0^+} \frac {\map f {x + h} - \map f x} h$ If the '''right-hand derivative''' exists, then $f$ is said to be '''right-hand differentiable''' at $x$.
Definition:Left-Hand Derivative/Real Function
Let $f: \R \to \R$ be a real function. The '''left-hand derivative''' of $f$ is defined as the left-hand limit: :$\displaystyle \map {f'_-} x = \lim_{h \mathop \to 0^-} \frac {\map f {x + h} - \map f x} h$ If the '''left-hand derivative''' exists, then $f$ is said to be '''left-hand differentiable''' at $x$.
Definition:Ideal Quotient
Let $A$ be a commutative ring with unity. Let $\mathfrak a, \mathfrak b \subseteq A$ be ideals of $A$. Their '''ideal quotient''' is the ideal consisting of elements whose product with $\mathfrak b$ is a subset of $\mathfrak a$: :$\ideal {\mathfrak a : \mathfrak b} := \set {x \in A : x \mathfrak b \subseteq \mathfrak a}$
Definition:Annihilator of Ideal of Ring
Let $A$ be a commutative ring with unity. Let $I \subseteq A$ be an ideal. === Definition 1 === The '''annihilator''' of $I$ is the ideal consisting of the elements $a \in A$ such that $a \cdot x = 0$ for all $x \in I$, where $0 \in A$ is its zero. === Definition 2 === The '''annihilator''' of $I$ is the ideal quotient $\ideal {0 : I}$, where $0$ is the zero ideal.
Definition:Extension of Ideal
Let $A$ and $B$ be commutative ring with unity. Let $f : A \to B$ be a ring homomorphism. Let $\mathfrak a \subseteq A$ be an ideal. The '''extension''' of $\mathfrak a$ by $f$ is the ideal generated by its image under $f$: :$\mathfrak a^e = \left\langle f(\mathfrak a) \right\rangle$
Definition:Irreducible Polynomial/Definition 1
Let $R$ be an integral domain. An '''irreducible polynomial''' over $R$ is an irreducible element of the polynomial ring $R \left[{X}\right]$.
Definition:Irreducible Polynomial/Definition 2
Let $K$ be a field. An '''irreducible polynomial''' over $K$ is a nonconstant polynomial over $K$ that is not the product of two polynomials of smaller degree.
Definition:Irreducible Polynomial/Definition 3
Let $K$ be a field. An '''irreducible polynomial''' over $K$ is a polynomial over $K$ that is not the product of two nonconstant polynomials.
Definition:Linear Ring Action
Let $R$ be a ring. Let $M$ be an abelian group. === Left Ring Action === {{:Definition:Linear Ring Action/Left}} === Right Ring Action === {{:Definition:Linear Ring Action/Right}}
Definition:Gradation on Abelian Group
Let $G$ be an abelian group. Let $\Delta$ be a set. A '''gradation of type $\Delta$''' on $G$ is a family of subgroups $\family {G_\lambda}_{\lambda \in \Delta}$ of which $G$ is the internal direct sum.
Definition:Graded Abelian Group
Let $\Delta$ be a set. A '''graded abelian group of type $\Delta$''' is a pair $\struct {G, f}$ where: :$G$ is an abelian group :$f$ is a gradation on $G$ indexed by $\Delta$, the '''set of degrees'''.
Definition:N-Graded Ring
Let $\N$ be the set of natural numbers. An '''$\N$-graded ring''' or '''positively $\Z$-graded ring''' is a graded ring of type the additive monoid of natural numbers. That is, it is a pair $\struct {R, f}$ where: :$R$ is a ring :$f$ is a sequence $\sequence {R_n}_{n \mathop \in \N}$ of subgroups of the additive group of $R$, of which it is the internal direct sum, and such that: ::$\forall x \in R_n, y \in R_m: x y \in R_{m + n}$
Definition:Z-Graded Ring
Let $\Z$ be the set of integers. A '''$\Z$-graded ring''' is a graded ring of type the additive group of integers. That is, it is a pair $\struct {R, f}$ where: :$R$ is a ring :$f$ is a family $\family {R_n}_{n \mathop \in \Z}$ of subgroups of the additive group of $R$, of which it is the internal direct sum, and such that: ::$\forall x \in R_n, y \in R_m: x y \in R_{m + n}$
Definition:Opposite Ring
Let $\struct {R, +, \times}$ be a ring. Let $* : R \times R \to R$ be the binary operation on $S$ defined by: :$\forall x, y \in S: x * y = y \times x$ The '''opposite ring''' of $R$ is the algebraic structure $\struct {R, +, *}$.
Definition:Ring Representation Defined by Ring Action
Let $R$ be a ring. Let $M$ be an abelian group. Let $\phi : R \times M \to M$ be a left linear ring action. The '''associated ring representation''' is the ring representation $\rho: R \to \map {\operatorname {End} } M$ with: :$\map {\map \rho r} m = \map \phi {r, m}$
Definition:Category of Modules
Let $R$ be a commutative ring. The '''category of $R$-modules''' is the category $\mathbf{R-Mod}$ with: {{DefineCategory |ob = $R$-modules |mor = $R$-module homomorphisms |id = identity mappings |comp = composition of mappings }} === Category of Unitary Modules === {{Definition:Category of Unitary Modules}}
Definition:Underlying Group of Module
Let $R$ be a ring. Let $\struct {M, +, \times}$ be a left module or right module over $R$. Its '''underlying (abelian) group''' is the abelian group $\struct {M, +}$.
Definition:Biadditive Mapping
Let $M, N, P$ be abelian groups. Let $M \times N$ be the cartesian product. A '''biadditive mapping''' $f : M \times N \to P$ is a mapping such that: :$\forall m_1, m_2 \in M : \forall n \in N: \map f {m_1 + m_2, n} = \map f {m_1, n} + \map f {m_2, n}$ :$\forall m \in M: \forall n_1, n_2 \in N: \map f {m, n_1 + n_2} = \map f {m, n_1} + \map f {m, n_2}$
Definition:Cartesian Square of Set
Let $S$ be a set. Its '''cartesian square''' is the cartesian power $S^2$.
Definition:R-Balanced Mapping
Let $R$ be a ring. Let $M$ be a right $R$-module and $N$ be a left $R$-module. Let $M \times N$ be their cartesian product. Let $P$ be an abelian group. An '''$R$-balanced mapping''' $f : M \times N \to P$ is a biadditive mapping with: :$\forall m \in M: \forall n \in N: \forall r \in R: \map f {m \cdot r, n} = \map f {m, r \cdot n}$
Definition:Tensor Product of Abelian Groups
Let $A$ and $B$ be abelian groups. === Definition 1: by universal property === Their '''tensor product''' is a pair $(A \otimes B, \theta)$ where: * $A \otimes B$ is an abelian group * $\theta : A \times B \to A \otimes B$ is a biadditive mapping such that, for every pair $(C, \omega)$ where: * $C$ is an abelian group * $\omega : A \times B \to C$ is a biadditive mapping there exists a unique group homomorphism $g : A \otimes B \to C$ with $\omega = g \circ \theta$. === Definition 2: construction === Their '''tensor product''' is the pair $(A \otimes B, \theta)$ where: * $A \otimes B$ is the quotient of the free abelian group $\Z^{(A \times B)}$ on the cartesian product $A \times B$ by the subgroup generated by the elements of the form: ** $(a_1 + a_2, b) - (a_1, b) - (a_2, b)$ ** $(a, b_1 + b_2) - (a, b_1) - (a, b_2)$ *:for $a, a_1, a_2 \in A$, $b, b_1, b_2 \in B$, where we denote $(a, b)$ for its image under the canonical mapping $A \times B \to \Z^{(A\times B)}$. * $\theta : A \times B \to A \otimes B$ is the composition of the canonical mapping $A \times B \to \Z^{(A\times B)}$ with the quotient group epimorphism $\Z^{(A\times B)} \to A \otimes B$.
Definition:Right Ring Action Defined by Ring Antirepresentation
Let $R$ be a ring. Let $M$ be an abelian group. Let $\rho: R \to \map {\operatorname {End} } M$ be a ring antirepresentation. The '''associated right ring action''' is the right linear ring action: :$M \times R \to M$: :$\tuple {m, r} \mapsto \map \rho r \paren m$