Rings - Revision history

Khanh: /* Category-theoretic description */

2021-12-19T22:43:36Z

Category-theoretic description

Khanh: /* Some examples of the ubiquity of rings */

2021-12-19T22:40:26Z

Some examples of the ubiquity of rings

Khanh: /* Rings with extra structure */

2021-12-19T22:33:42Z

Rings with extra structure

Khanh: /* Special kinds of rings */

2021-12-19T22:29:07Z

Special kinds of rings

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Khanh: /* Constructions */

2021-12-19T22:03:23Z

Constructions

Khanh: /* Constructions */

2021-12-19T21:58:41Z

Constructions

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Khanh: /* Module */

2021-12-19T21:44:50Z

Module

Khanh: /* Homomorphism */

2021-12-19T21:41:35Z

Homomorphism

Khanh: /* Basic concepts */

2021-12-19T21:38:35Z

Basic concepts

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Khanh: /* Noncommutative rings */

2021-12-19T21:04:27Z

Noncommutative rings

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 == Category-theoretic description ==
-Every ring can be thought of as a [[monoid (category theory)|monoid]] in '''Ab''', the [[category of abelian groups]] (thought of as a [[monoidal category]] under the [[tensor product of abelian groups|tensor product of <math>{\mathbf Z}</math>-modules]]). The monoid action of a ring ''R'' on an abelian group is simply an [[module (mathematics)|''R''-module]]. Essentially, an ''R''-module is a generalization of the notion of a [[vector space]] – where rather than a vector space over a field, one has a "vector space over a ring".
+Every ring can be thought of as a monoid in '''Ab''', the category of abelian groups (thought of as a monoidal category under the tensor product of <math>{\mathbf Z}</math>-modules). The monoid action of a ring ''R'' on an abelian group is simply an ''R''-module. Essentially, an ''R''-module is a generalization of the notion of a vector space – where rather than a vector space over a field, one has a "vector space over a ring".
-Let {{nowrap|(''A'', +)}} be an abelian group and let End(''A'') be its [[endomorphism ring]] (see above). Note that, essentially, End(''A'') is the set of all morphisms of ''A'', where if ''f'' is in End(''A''), and ''g'' is in End(''A''), the following rules may be used to compute {{nowrap|''f'' + ''g''}} and {{nowrap|''f'' '''⋅''' ''g''}}:
+Let (''A'', +) be an abelian group and let End(''A'') be its endomorphism ring (see above). Note that, essentially, End(''A'') is the set of all morphisms of ''A'', where if ''f'' is in End(''A''), and ''g'' is in End(''A''), the following rules may be used to compute ''f'' + ''g'' and ''f'' '''⋅''' ''g'':
 * (''f'' +&thinsp;''g'')(''x'') = ''f''(''x'')&thinsp;+&thinsp;''g''(''x'')
 * (''f'' '''⋅'''&thinsp;''g'')(''x'') = ''f''(''g''(''x'')),
-where + as in {{nowrap|''f''(''x'') + ''g''(''x'')}} is addition in ''A'', and function composition is denoted from right to left. Therefore, [[functor|associated]] to any abelian group, is a ring. Conversely, given any ring, {{nowrap|(''R'', +, '''⋅''' )}}, {{nowrap|(''R'', +)}} is an abelian group. Furthermore, for every ''r'' in ''R'', right (or left) multiplication by ''r'' gives rise to a morphism of {{nowrap|(''R'', +)}}, by right (or left) distributivity. Let {{nowrap|1=''A'' = (''R'', +)}}. Consider those [[endomorphism]]s of ''A'', that "factor through" right (or left) multiplication of ''R''. In other words, let End<sub>''R''</sub>(''A'') be the set of all morphisms ''m'' of ''A'', having the property that {{nowrap|1=''m''(''r'' '''⋅'''&thinsp;''x'') = ''r'' '''⋅''' ''m''(''x'')}}. It was seen that every ''r'' in ''R'' gives rise to a morphism of ''A'': right multiplication by ''r''. It is in fact true that this association of any element of ''R'', to a morphism of ''A'', as a function from ''R'' to End<sub>''R''</sub>(''A''), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian ''X''-group (by ''X''-group, it is meant a group with ''X'' being its [[Group with operators|set of operators]]). In essence, the most general form of a ring, is the endomorphism group of some abelian ''X''-group.
+where + as in ''f''(''x'') + ''g''(''x'') is addition in ''A'', and function composition is denoted from right to left. Therefore, associated to any abelian group, is a ring. Conversely, given any ring, (''R'', +, '''⋅''' ), (''R'', +) is an abelian group. Furthermore, for every ''r'' in ''R'', right (or left) multiplication by ''r'' gives rise to a morphism of (''R'', +), by right (or left) distributivity. Let ''A'' = (''R'', +). Consider those endomorphisms of ''A'', that "factor through" right (or left) multiplication of ''R''. In other words, let End<sub>''R''</sub>(''A'') be the set of all morphisms ''m'' of ''A'', having the property that ''m''(''r'' '''⋅'''&thinsp;''x'') = ''r'' '''⋅''' ''m''(''x''). It was seen that every ''r'' in ''R'' gives rise to a morphism of ''A'': right multiplication by ''r''. It is in fact true that this association of any element of ''R'', to a morphism of ''A'', as a function from ''R'' to End<sub>''R''</sub>(''A''), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian ''X''-group (by ''X''-group, it is meant a group with ''X'' being its set of operators). In essence, the most general form of a ring, is the endomorphism group of some abelian ''X''-group.
-Any ring can be seen as a [[preadditive category]] with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. [[Additive functor]]s between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of [[morphism]]s closed under addition and under composition with arbitrary morphisms.
+Any ring can be seen as a preadditive category with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms.
 == Generalization ==

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 == Some examples of the ubiquity of rings ==
-Many different kinds of [[mathematical object]]s can be fruitfully analyzed in terms of some [[functor|associated ring]].
+Many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring.
 === Cohomology ring of a topological space ===
-To any [[topological space]] ''X'' one can associate its integral [[cohomology ring]]
+To any topological space ''X'' one can associate its integral cohomology ring
 :<math>H^*(X,\mathbf{Z}) = \bigoplus_{i=0}^{\infty} H^i(X,\mathbf{Z}),</math>
-a [[graded ring]]. There are also [[homology group]]s <math>H_i(X,\mathbf{Z})</math> of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the [[sphere]]s and [[torus|tori]], for which the methods of [[point-set topology]] are not well-suited. [[Cohomology group]]s were later defined in terms of homology groups in a way which is roughly analogous to the dual of a [[vector space]]. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the [[universal coefficient theorem]]. However, the advantage of the cohomology groups is that there is a [[cup product|natural product]], which is analogous to the observation that one can multiply pointwise a ''k''-[[multilinear form]] and an ''l''-multilinear form to get a ({{nowrap|''k'' + ''l''}})-multilinear form.
+a graded ring. There are also homology groups <math>H_i(X,\mathbf{Z})</math> of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the spheres and tori, for which the methods of point-set topology are not well-suited. Cohomology groups were later defined in terms of homology groups in a way which is roughly analogous to the dual of a vector space. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the universal coefficient theorem. However, the advantage of the cohomology groups is that there is a natural product, which is analogous to the observation that one can multiply pointwise a ''k''-multilinear form and an ''l''-multilinear form to get a (''k'' + ''l'')-multilinear form.
-The ring structure in cohomology provides the foundation for [[characteristic class]]es of [[fiber bundle]]s, intersection theory on manifolds and [[algebraic variety|algebraic varieties]], [[Schubert calculus]] and much more.
+The ring structure in cohomology provides the foundation for characteristic classes of fiber bundles, intersection theory on manifolds and algebraic varieties, Schubert calculus and much more.
 === Burnside ring of a group ===
-To any [[group (mathematics)|group]] is associated its [[Burnside ring]] which uses a ring to describe the various ways the group can [[Group action (mathematics)|act]] on a finite set. The Burnside ring's additive group is the [[free abelian group]] whose basis are the transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the [[representation ring]]: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.
+To any group is associated its Burnside ring which uses a ring to describe the various ways the group can act on a finite set. The Burnside ring's additive group is the free abelian group whose basis are the transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the representation ring: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.
 === Representation ring of a group ring ===
-To any [[group ring]] or [[Hopf algebra]] is associated its [[representation ring]] or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from [[character theory]], which is more or less the [[Grothendieck group]] given a ring structure.
+To any group ring or Hopf algebra is associated its representation ring or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from character theory, which is more or less the Grothendieck group given a ring structure.
 === Function field of an irreducible algebraic variety ===
-To any irreducible [[algebraic variety]] is associated its [[function field of an algebraic variety|function field]]. The points of an algebraic variety correspond to [[valuation ring]]s contained in the function field and containing the [[coordinate ring]]. The study of [[algebraic geometry]] makes heavy use of [[commutative algebra]] to study geometric concepts in terms of ring-theoretic properties. [[Birational geometry]] studies maps between the subrings of the function field.
+To any irreducible algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. The study of algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ring-theoretic properties. Birational geometry studies maps between the subrings of the function field.
 === Face ring of a simplicial complex ===
-Every [[simplicial complex]] has an associated face ring, also called its [[Stanley–Reisner ring]]. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in [[algebraic combinatorics]]. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of [[simplicial polytope]]s.
+Every simplicial complex has an associated face ring, also called its Stanley–Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in algebraic combinatorics. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of simplicial polytopes.
 == Category-theoretic description ==

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 == Rings with extra structure ==
-A ring may be viewed as an [[abelian group]] (by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:
+A ring may be viewed as an abelian group (by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:
-* An [[associative algebra]] is a ring that is also a [[vector space]] over a field ''K'' such that the scalar multiplication is compatible with the ring multiplication. For instance, the set of ''n''-by-''n'' matrices over the real field '''R''' has dimension ''n''<sup>2</sup> as a real vector space.
+* An associative algebra is a ring that is also a vector space over a field ''K'' such that the scalar multiplication is compatible with the ring multiplication. For instance, the set of ''n''-by-''n'' matrices over the real field '''R''' has dimension ''n''<sup>2</sup> as a real vector space.
-* A ring ''R'' is a [[topological ring]] if its set of elements ''R'' is given a [[topological space|topology]] which makes the addition map ( <math>+ : R\times R \to R\,</math>) and the multiplication map ( <math>\cdot : R\times R \to R\,</math>) to be both [[Continuous function (topology)|continuous]] as maps between topological spaces (where ''X'' × ''X'' inherits the [[product topology]] or any other product in the category). For example, ''n''-by-''n'' matrices over the real numbers could be given either the [[Euclidean topology]], or the [[Zariski topology]], and in either case one would obtain a topological ring.
+* A ring ''R'' is a topological ring if its set of elements ''R'' is given a topology which makes the addition map ( <math>+ : R\times R \to R\,</math>) and the multiplication map ( <math>\cdot : R\times R \to R\,</math>) to be both continuous as maps between topological spaces (where ''X'' × ''X'' inherits the product topology or any other product in the category). For example, ''n''-by-''n'' matrices over the real numbers could be given either the Euclidean topology, or the Zariski topology, and in either case one would obtain a topological ring.
-* A [[λ-ring]] is a commutative ring ''R'' together with operations {{nowrap|λ<sup>''n''</sup>: ''R'' → ''R''}} that are like ''n''-th [[exterior power]]s:
+* A λ-ring is a commutative ring ''R'' together with operations λ<sup>''n''</sup>: ''R'' → ''R'' that are like ''n''-th exterior powers:
 ::<math>\lambda^n(x + y) = \sum_0^n \lambda^i(x) \lambda^{n-i}(y)</math>.
-:For example, '''Z''' is a λ-ring with <math>\lambda^n(x) = \binom{x}{n}</math>, the [[binomial coefficient]]s. The notion plays a central rule in the algebraic approach to the [[Riemann–Roch theorem]].
+:For example, '''Z''' is a λ-ring with <math>\lambda^n(x) = \binom{x}{n}</math>, the binomial coefficients. The notion plays a central rule in the algebraic approach to the Riemann–Roch theorem.
-* A [[totally ordered ring]] is a ring with a [[total ordering]] that is compatible with ring operations.<!-- Z is characterized as a certain unique ordered ring. Can’t remember the precise statement.-->
+* A totally ordered ring is a ring with a total ordering that is compatible with ring operations.
 == Some examples of the ubiquity of rings ==

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 Let ''k'' be an algebraically closed field. The Hilbert's Nullstellensatz (theorem of zeros) states that there is a natural one-to-one correspondence between the set of all prime ideals in <math>k\left[t_1, \ldots, t_n\right]</math> and the set of closed subvarieties of <math>k^n</math>. In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring. (cf. Gröbner basis.)
-There are some other related constructions. A [[formal power series ring]] <math>R[\![t]\!]</math> consists of formal power series
+There are some other related constructions. A formal power series ring <math>R[\![t]\!]</math> consists of formal power series
 : <math>\sum_0^\infty a_i t^i, \quad a_i \in R</math>
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 The localization generalizes the construction of the field of fractions of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring ''R'' and a subset ''S'' of ''R'', there exists a ring <math>R[S^{-1}]</math> together with the ring homomorphism <math>R \to R\left[S^{-1}\right]</math> that "inverts" ''S''; that is, the homomorphism maps elements in ''S'' to unit elements in <math>R\left[S^{-1}\right]</math>, and, moreover, any ring homomorphism from ''R'' that "inverts" ''S'' uniquely factors through <math>R\left[S^{-1}\right]</math>. The ring <math>R\left[S^{-1}\right]</math> is called the '''localization''' of ''R'' with respect to ''S''. For example, if ''R'' is a commutative ring and ''f'' an element in ''R'', then the localization <math>R\left[f^{-1}\right]</math> consists of elements of the form <math>r/f^n, \, r \in R , \, n \ge 0</math> (to be precise, <math>R\left[f^{-1}\right] = R[t]/(tf - 1).</math>)
-The localization is frequently applied to a commutative ring ''R'' with respect to the complement of a prime ideal (or a union of prime ideals) in&nbsp;''R''. In that case <math>S = R - \mathfrak{p}</math>, one often writes <math>R_\mathfrak{p}</math> for <math>R\left[S^{-1}\right]</math>. <math>R_\mathfrak{p}</math> is then a [[local ring]] with the [[maximal ideal]] <math>\mathfrak{p} R_\mathfrak{p}</math>. This is the reason for the terminology "localization". The field of fractions of an integral domain ''R'' is the localization of ''R'' at the prime ideal zero. If <math>\mathfrak{p}</math> is a prime ideal of a commutative ring&nbsp;''R'', then the field of fractions of <math>R/\mathfrak{p}</math> is the same as the residue field of the local ring <math>R_\mathfrak{p}</math> and is denoted by <math>k(\mathfrak{p})</math>.
+The localization is frequently applied to a commutative ring ''R'' with respect to the complement of a prime ideal (or a union of prime ideals) in&nbsp;''R''. In that case <math>S = R - \mathfrak{p}</math>, one often writes <math>R_\mathfrak{p}</math> for <math>R\left[S^{-1}\right]</math>. <math>R_\mathfrak{p}</math> is then a local ring with the maximal ideal <math>\mathfrak{p} R_\mathfrak{p}</math>. This is the reason for the terminology "localization". The field of fractions of an integral domain ''R'' is the localization of ''R'' at the prime ideal zero. If <math>\mathfrak{p}</math> is a prime ideal of a commutative ring&nbsp;''R'', then the field of fractions of <math>R/\mathfrak{p}</math> is the same as the residue field of the local ring <math>R_\mathfrak{p}</math> and is denoted by <math>k(\mathfrak{p})</math>.
 If ''M'' is a left ''R''-module, then the localization of ''M'' with respect to ''S'' is given by a change of rings <math>M\left[S^{-1}\right] = R\left[S^{-1}\right] \otimes_R M</math>.

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 == Module ==
-The concept of a ''module over a ring'' generalizes the concept of a [[vector space]] (over a [[field (mathematics)|field]]) by generalizing from multiplication of vectors with elements of a field ([[scalar multiplication]]) to multiplication with elements of a ring. More precisely, given a ring {{math|''R''}} with&nbsp;1, an {{math|''R''}}-module {{math|''M''}} is an [[abelian group]] equipped with an [[operation (mathematics)|operation]] {{math|''R'' × ''M'' → ''M''}} (associating an element of {{math|''M''}} to every pair of an element of {{math|''R''}} and an element of {{math|''M''}}) that satisfies certain [[axiom#Non-logical axioms|axioms]]. This operation is commonly denoted multiplicatively and called multiplication. The axioms of modules are the following: for all {{math|''a'', ''b''}} in {{math|''R''}} and all {{math|''x'', ''y''}} in {{math|''M''}}, we have:
+The concept of a ''module over a ring'' generalizes the concept of a vector space (over a field) by generalizing from multiplication of vectors with elements of a field (scalar multiplication) to multiplication with elements of a ring. More precisely, given a ring {{math|''R''}} with&nbsp;1, an {{math|''R''}}-module {{math|''M''}} is an abelian group equipped with an operation {{math|''R'' × ''M'' → ''M''}} (associating an element of {{math|''M''}} to every pair of an element of {{math|''R''}} and an element of {{math|''M''}}) that satisfies certain axioms. This operation is commonly denoted multiplicatively and called multiplication. The axioms of modules are the following: for all {{math|''a'', ''b''}} in {{math|''R''}} and all {{math|''x'', ''y''}} in {{math|''M''}}, we have:
 * {{math|''M''}} is an abelian group under addition.
 * <math>a(x+y)=ax+ay</math>
@@ Line 220: / Line 219: @@
 * <math>1x=x</math>
 * <math>(ab)x=a(bx)</math>
-When the ring is [[noncommutative ring|noncommutative]] these axioms define ''left modules''; ''right modules'' are defined similarly by writing {{math|''xa''}} instead of {{math|''ax''}}. This is not only a change of notation, as the last axiom of right modules (that is {{math|1=''x''(''ab'') = (''xa'')''b''}}) becomes {{math|1=(''ab'')''x'' = ''b''(''ax'')}}, if left multiplication (by ring elements) is used for a right module.
+When the ring is noncommutative these axioms define ''left modules''; ''right modules'' are defined similarly by writing {{math|''xa''}} instead of {{math|''ax''}}. This is not only a change of notation, as the last axiom of right modules (that is {{math|1=''x''(''ab'') = (''xa'')''b''}}) becomes {{math|1=(''ab'')''x'' = ''b''(''ax'')}}, if left multiplication (by ring elements) is used for a right module.
 Basic examples of modules are ideals, including the ring itself.
-Although similarly defined, the theory of modules is much more complicated than that of vector space, mainly, because, unlike vector spaces, modules are not characterized (up to an isomorphism) by a single invariant (the [[dimension (vector space)|dimension of a vector space]]). In particular, not all modules have a [[basis (linear algebra)|basis]].
+Although similarly defined, the theory of modules is much more complicated than that of vector space, mainly, because, unlike vector spaces, modules are not characterized (up to an isomorphism) by a single invariant (the dimension of a vector space). In particular, not all modules have a basis.
-The axioms of modules imply that {{math|1=(−1)''x'' = −''x''}}, where the first minus denotes the [[additive inverse]] in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.
+The axioms of modules imply that {{math|1=(−1)''x'' = −''x''}}, where the first minus denotes the additive inverse in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.
-Any ring homomorphism induces a structure of a module: if {{math|''f'' : ''R'' → ''S''}} is a ring homomorphism, then {{math|''S''}} is a left module over {{math|''R''}} by the multiplication: {{nowrap|1=''rs'' = ''f''(''r'')''s''}}. If {{math|''R''}} is commutative or if {{math|''f''(''R'')}} is contained in the [[center of a ring|center]] of {{math|''S''}}, the ring {{math|''S''}} is called a {{math|''R''}}-[[algebra over a ring|algebra]]. In particular, every ring is an algebra over the integers.
+Any ring homomorphism induces a structure of a module: if {{math|''f'' : ''R'' → ''S''}} is a ring homomorphism, then {{math|''S''}} is a left module over {{math|''R''}} by the multiplication: ''rs'' = ''f''(''r'')''s''. If {{math|''R''}} is commutative or if {{math|''f''(''R'')}} is contained in the center of {{math|''S''}}, the ring {{math|''S''}} is called a {{math|''R''}}-algebra. In particular, every ring is an algebra over the integers.
 == Constructions ==

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 Examples:
-* The function that maps each integer ''x'' to its remainder modulo 4 (a number in {{mset|0, 1, 2, 3}}) is a homomorphism from the ring '''Z''' to the quotient ring '''Z'''/4'''Z''' ("quotient ring" is defined below).
+* The function that maps each integer ''x'' to its remainder modulo 4 (a number in {0, 1, 2, 3}) is a homomorphism from the ring '''Z''' to the quotient ring '''Z'''/4'''Z''' ("quotient ring" is defined below).
 * If <math>u</math> is a unit element in a ring ''R'', then <math>R \to R, x \mapsto uxu^{-1}</math> is a ring homomorphism, called an inner automorphism of ''R''.
 * Let ''R'' be a commutative ring of prime characteristic ''p''. Then <math>x \mapsto x^p</math> is a ring endomorphism of ''R'' called the Frobenius homomorphism.
@@ Line 196: / Line 196: @@
 Given a ring homomorphism <math>f:R \to S</math>, the set of all elements mapped to 0 by ''f'' is called the kernel of&nbsp;''f''. The kernel is a two-sided ideal of&nbsp;''R''. The image of&nbsp;''f'', on the other hand, is not always an ideal, but it is always a subring of&nbsp;''S''.
-To give a ring homomorphism from a commutative ring ''R'' to a ring ''A'' with image contained in the center of ''A'' is the same as to give a structure of an [[associative algebra|algebra]] over ''R'' to&nbsp;''A'' (which in particular gives a structure of an ''A''-module).
+To give a ring homomorphism from a commutative ring ''R'' to a ring ''A'' with image contained in the center of ''A'' is the same as to give a structure of an algebra over ''R'' to&nbsp;''A'' (which in particular gives a structure of an ''A''-module).
 === Quotient ring ===

@@ Line 119: / Line 119: @@
 === Noncommutative rings ===
-* For any ring ''R'' and any natural number ''n'', the set of all square ''n''-by-''n'' matrices with entries from ''R'', forms a ring with matrix addition and matrix multiplication as operations. For {{nowrap|1=''n'' = 1}}, this matrix ring is isomorphic to ''R'' itself. For ''n'' > 1 (and ''R'' not the zero ring), this matrix ring is noncommutative.
+* For any ring ''R'' and any natural number ''n'', the set of all square ''n''-by-''n'' matrices with entries from ''R'', forms a ring with matrix addition and matrix multiplication as operations. For ''n'' = 1, this matrix ring is isomorphic to ''R'' itself. For ''n'' > 1 (and ''R'' not the zero ring), this matrix ring is noncommutative.
 * If ''G'' is an abelian group, then the endomorphisms of ''G'' form a ring, the endomorphism ring End(''G'') of&nbsp;''G''. The operations in this ring are addition and composition of endomorphisms. More generally, if ''V'' is a left module over a ring ''R'', then the set of all ''R''-linear maps forms a ring, also called the endomorphism ring and denoted by End<sub>''R''</sub>(''V'').
 * If ''G'' is a group and ''R'' is a ring, the group ring of ''G'' over ''R'' is a free module over ''R'' having ''G'' as basis. Multiplication is defined by the rules that the elements of ''G'' commute with the elements of ''R'' and multiply together as they do in the group ''G''.