Editing Endomorphism ring (section)

== 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]].