Editing Matrix ring (section)

== Structure ==
* The matrix ring M<sub>''n''</sub>(''R'') can be identified with the [[ring of endomorphisms]] of the [[free module|free right ''R''-module]] of rank ''n''; that is, {{nowrap|M<sub>''n''</sub>(''R'') ≅ End<sub>''R''</sub>(''R''<sup>''n''</sup>)}}.  [[Matrix multiplication]] corresponds to composition of endomorphisms.
* The ring M<sub>''n''</sub>(''D'') over a [[division ring]] ''D'' is an [[Artinian ring|Artinian]] [[simple ring]], a special type of [[semisimple ring]].  The rings <math>\mathbb{CFM}_I(D)</math> and <math>\mathbb{RFM}_I(D)</math> are ''not'' simple and not Artinian if the set ''I'' is infinite, but they are still [[full linear ring]]s.
* The [[Artin–Wedderburn theorem]] states that every semisimple ring is isomorphic to a finite [[direct product]] <math display="inline">\prod_{i=1}^r \operatorname{M}_{n_i}(D_i)</math>, for some nonnegative integer ''r'', positive integers ''n''<sub>''i''</sub>, and division rings ''D''<sub>''i''</sub>.
* When we view M<sub>''n''</sub>('''C''') as the ring of linear endomorphisms of '''C'''<sup>''n''</sup>, those matrices which vanish on a given subspace ''V'' form a [[left ideal]]. Conversely, for a given left ideal ''I'' of M<sub>''n''</sub>('''C''') the intersection of [[Kernel (linear algebra)|null spaces]] of all matrices in ''I'' gives a subspace of '''C'''<sup>''n''</sup>. Under this construction, the left ideals of M<sub>''n''</sub>('''C''') are in bijection with the subspaces of '''C'''<sup>''n''</sup>.
* There is a bijection between the two-sided [[ideal (ring theory)|ideals]] of M<sub>''n''</sub>(''R'') and the two-sided ideals of ''R''.  Namely, for each ideal ''I'' of ''R'', the set of all {{nowrap|''n'' × ''n''}} matrices with entries in ''I'' is an ideal of M<sub>''n''</sub>(''R''), and each ideal of M<sub>''n''</sub>(''R'') arises in this way.  This implies that M<sub>''n''</sub>(''R'') is [[simple ring|simple]] if and only if ''R'' is simple. For {{nowrap|''n'' ≥ 2}}, not every left ideal or right ideal of M<sub>''n''</sub>(''R'') arises by the previous construction from a left ideal or a right ideal in ''R''. For example, the set of matrices whose columns with indices 2 through ''n'' are all zero forms a left ideal in M<sub>''n''</sub>(''R'').
* The previous ideal correspondence actually arises from the fact that the rings ''R'' and M<sub>''n''</sub>(''R'') are [[Morita equivalent]]. Roughly speaking, this means that the category of left ''R''-modules and the category of left M<sub>''n''</sub>(''R'')-modules are very similar. Because of this, there is a natural bijective correspondence between the ''isomorphism classes'' of left ''R''-modules and left M<sub>''n''</sub>(''R'')-modules, and between the isomorphism classes of left ideals of ''R'' and left ideals of M<sub>''n''</sub>(''R''). Identical statements hold for right modules and right ideals.  Through Morita equivalence, M<sub>''n''</sub>(''R'') inherits any [[Morita equivalence|Morita-invariant]] properties of ''R'', such as being [[simple ring|simple]], [[Artinian ring|Artinian]], [[Noetherian ring|Noetherian]], [[Prime ring|prime]].