Editing Ring (mathematics) (section)

=== Matrix ring and endomorphism ring ===
{{Main|Matrix ring|Endomorphism ring}}

Let {{mvar|R}} be a ring (not necessarily commutative). The set of all square matrices of size {{mvar|n}} with entries in {{mvar|R}} forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the [[matrix ring]] and is denoted by {{math|M{{sub|''n''}}(''R'')}}. Given a right {{mvar|R}}-module {{mvar|U}}, the set of all {{mvar|R}}-linear maps from {{mvar|U}} to itself forms a ring with addition that is of function and multiplication that is of [[composition of functions]]; it is called the endomorphism ring of {{mvar|U}} and is denoted by {{math|End{{sub|''R''}}(''U'')}}.

As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: <math>\operatorname{End}_R(R^n) \simeq \operatorname{M}_n(R).</math> This is a special case of the following fact: If <math>f: \oplus_1^n U \to \oplus_1^n U</math> is an {{mvar|R}}-linear map, then {{mvar|f}} may be written as a matrix with entries {{mvar|f{{sub|ij}}}} in {{math|1=''S'' = End{{sub|''R''}}(''U'')}}, resulting in the ring isomorphism:
:<math>\operatorname{End}_R(\oplus_1^n U) \to \operatorname{M}_n(S), \quad f \mapsto (f_{ij}).</math>

Any ring homomorphism {{math|''R'' → ''S''}} induces {{math|M{{sub|''n''}}(''R'') → M{{sub|''n''}}(''S'')}}.{{sfnp|Cohn|2003|loc=4.4|ps=}}

[[Schur's lemma]] says that if {{mvar|U}} is a simple right {{mvar|R}}-module, then {{math|End{{sub|''R''}}(''U'')}} is a division ring.{{sfnp|Lang|2002|loc=Ch. XVII. Proposition 1.1|ps=}} If <math>U = \bigoplus_{i = 1}^r U_i^{\oplus m_i}</math> is a direct sum of {{mvar|m{{sub|i}}}}-copies of simple {{mvar|R}}-modules <math>U_i,</math> then
:<math>\operatorname{End}_R(U) \simeq \prod_{i=1}^r \operatorname{M}_{m_i} (\operatorname{End}_R(U_i)).</math>
The [[Artin–Wedderburn theorem]] states any [[semisimple ring]] (cf. below) is of this form.

A ring {{mvar|R}} and the matrix ring {{math|M{{sub|''n''}}(''R'')}} over it are [[Morita equivalent]]: the [[Category (mathematics)|category]] of right modules of {{mvar|R}} is equivalent to the category of right modules over {{math|M{{sub|''n''}}(''R'')}}.{{sfnp|Cohn|2003|loc=4.4|ps=}} In particular, two-sided ideals in {{mvar|R}} correspond in one-to-one to two-sided ideals in {{math|M{{sub|''n''}}(''R'')}}.