Editing Endomorphism ring

{{Short description|Endomorphism algebra of an abelian group}}
In [[mathematics]], the [[endomorphism]]s of an [[abelian group]] ''X'' form a [[ring (mathematics)|ring]]. This ring is called the '''endomorphism ring''' of ''X'', denoted by End(''X''); the set of all [[homomorphism]]s of ''X'' into itself. Addition of endomorphisms arises naturally in a [[Pointwise#Pointwise operations|pointwise]] manner and multiplication via [[function composition|endomorphism composition]].  Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the [[zero morphism|zero map]] <math display="inline">0: x \mapsto 0</math>  as [[additive identity]] and the [[identity map]] <math display="inline">1: x \mapsto x</math> as [[Identity element|multiplicative identity]].{{sfn|ps=none|Fraleigh|1976|p=211}}{{sfn|ps=none|Passman|1991|pp=4–5}}

The functions involved are restricted to what is defined as a homomorphism in the context, which depends upon the [[category (mathematics)|category]] of the object under consideration. The endomorphism ring consequently encodes several internal properties of the object. As the endomorphism ring is often an [[algebra (ring theory)|algebra]] over some ring ''R,'' this may also be called the '''endomorphism algebra'''.

An abelian group is the same thing as a [[Module (mathematics)|module]] over the ring of [[integer]]s, which is the [[initial object]] in the [[category of rings]].  In a similar fashion, if ''R'' is any [[commutative ring]], the endomorphisms of an ''R''-module form an [[algebra over a ring|algebra over ''R'']] by the same axioms and derivation.  In particular, if ''R'' is a [[Field (mathematics)|field]], its modules ''M'' are [[Vector space|vector spaces]] and the endomorphism ring of each is an [[Algebra over a field|algebra over the field]] ''R''.

== Description ==
Let {{nowrap|(''A'', +)}} be an abelian group and we consider the group homomorphisms from ''A'' into ''A''. Then addition of two such homomorphisms may be defined pointwise to produce another group homomorphism. Explicitly, given two such homomorphisms ''f'' and ''g'', the sum of ''f'' and ''g'' is the homomorphism {{nowrap|''f'' + ''g'' : ''x'' ↦ ''f''(''x'') + ''g''(''x'')}}. Under this operation End(''A'') is an abelian group. With the additional operation of composition of homomorphisms, End(''A'') is a ring with multiplicative identity. This composition is explicitly {{nowrap|''fg'' : ''x'' ↦ ''f''(''g''(''x''))}}. The multiplicative identity is the identity homomorphism on ''A''. The additive inverses are the pointwise inverses. 

If the set ''A'' does not form an ''abelian'' group, then the above construction is not necessarily well-defined, as then the sum of two homomorphisms need not be a homomorphism.{{sfn|ps=none|Dummit|Foote|p=347}} However, the closure of the set of endomorphisms under the above operations is a canonical example of a [[near-ring]] that is not a ring.

== Properties ==
* Endomorphism rings always have additive and multiplicative [[identity element|identities]], respectively the [[zero map]] and [[identity function|identity map]].
* Endomorphism rings are [[associative]], but typically [[non-commutative ring|non-commutative]].
* If a module is [[simple module|simple]], then its endomorphism ring is a [[division ring]] (this is sometimes called [[Schur's lemma]]).{{sfn|ps=none|Jacobson|2009|loc=p. 118}}
* A module is [[indecomposable module|indecomposable]] if and only if its endomorphism ring does not contain any non-trivial [[idempotent element (ring theory)|idempotent element]]s.{{sfn|ps=none|Jacobson|2009|loc=p. 111, Prop. 3.1}} If the module is an [[injective module]], then indecomposability is equivalent to the endomorphism ring being a [[local ring]].{{sfn|ps=none|Wisbauer|1991|loc=p. 163}}
* For a [[semisimple module]], the endomorphism ring is a [[von Neumann regular ring]].
* The endomorphism ring of a nonzero right [[uniserial module]] has either one or two maximal right ideals. If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring.
* The endomorphism ring of an Artinian [[uniform module]] is a local ring.{{sfn|ps=none|Wisbauer|1991|loc=p. 263}}
* The endomorphism ring of a module with finite [[composition length]] is a [[semiprimary ring]].
* The endomorphism ring of a [[continuous module]] or [[discrete module]] is a [[clean ring]].{{sfn|ps=none|Camillo|Khurana|Lam|Nicholson|2006}}
* If an ''R'' module is finitely generated and projective (that is, a [[progenerator]]), then the endomorphism ring of the module and ''R'' share all Morita invariant properties. A fundamental result of Morita theory is that all rings equivalent to ''R'' arise as endomorphism rings of progenerators.

== Examples ==
* In the category of ''R''-[[module (mathematics)|module]]s, the endomorphism ring of an ''R''-module ''M'' will only use the ''R''-[[module homomorphism]]s, which are typically a proper subset of the abelian group homomorphisms.{{refn|Abelian groups may also be viewed as modules over the ring of integers.}} When ''M'' is a [[finitely generated module|finitely generated]] [[projective module]], the endomorphism ring is central to [[Morita equivalence]] of module categories.
* For any abelian group <math>A</math>, <math>\mathrm{M}_n(\operatorname{End}(A))\cong \operatorname{End}(A^n)</math>, since any matrix in <math>\mathrm{M}_n(\operatorname{End}(A))</math> carries a natural homomorphism structure of <math>A^n</math> as follows: 
*: <math>\begin{pmatrix}\varphi_{11}&\cdots &\varphi_{1n}\\ \vdots& &\vdots \\ \varphi_{n1}&\cdots& \varphi_{nn}  \end{pmatrix}\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix}=\begin{pmatrix}\sum_{i=1}^n\varphi_{1i}(a_i)\\\vdots\\\sum_{i=1}^n\varphi_{ni}(a_i) \end{pmatrix}. </math>
: One can use this isomorphism to construct many non-commutative endomorphism rings. For example: <math>\operatorname{End}(\mathbb{Z}\times \mathbb{Z})\cong \mathrm{M}_2(\mathbb{Z})</math>, since <math>\operatorname{End}(\mathbb{Z})\cong \mathbb{Z}</math>.  
: Also, when <math>R=K</math> is a field, there is a canonical isomorphism <math>\operatorname{End}(K)\cong K</math>, so <math>\operatorname{End}(K^n)\cong \mathrm{M}_n(K)</math>, that is, the endomorphism ring of a <math>K</math>-vector space is identified with the [[matrix ring|ring of ''n''-by-''n'' matrices]] with entries in <math>K</math>.{{sfn|ps=none|Drozd|Kirichenko|1994|loc=pp. 23–31}} More generally, the endomorphism algebra of the [[free module]] <math>M = R^n</math> is naturally <math>n</math>-by-<math>n</math> matrices with entries in the ring <math>R</math>.
* As a particular example of the last point, for any ring ''R'' with unity, {{nowrap|1=End(''R''<sub>''R''</sub>) = ''R''}}, where the elements of ''R'' act on ''R'' by ''left'' multiplication.
* In general, endomorphism rings can be defined for the objects of any [[preadditive category]].

== Notes ==
<references/>

== References ==
{{refbegin}}
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}}
* {{citation
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}}
* {{citation
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}}
* {{springer|title=Endomorphism ring|id=p/e035610}}
* {{citation
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* {{citation
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* {{citation
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{{refend}}

[[Category:Ring theory]]
[[Category:Module theory]]
[[Category:Category theory]]