Editing Matrix ring (section)

{{Short description|Mathematical ring whose elements are matrices}}
{{Redirect|Matrix algebra|the algebraic theory of matrices|Matrix (mathematics)|and|Linear algebra}}

In [[abstract algebra]], a '''matrix ring''' is a set of [[matrix (mathematics)|matrices]] with entries in a [[ring (mathematics)|ring]] ''R'' that form a ring under [[matrix addition]] and [[matrix multiplication]].{{sfnp|Lam|1999|loc=Theorem 3.1|ps=}} The set of all {{nowrap|''n'' × ''n''}} matrices with entries in ''R'' is a matrix ring denoted M<sub>''n''</sub>(''R''){{sfnp|Lam|2001|}}{{sfnp|Lang|2005|loc=V.§3|ps=}}{{sfnp|Serre|2006|p=3|ps=}}{{sfnp|Serre|1979|p=158|ps=}} (alternative notations: Mat<sub>''n''</sub>(''R''){{sfnp|Lang|2005|loc=V.§3|ps=}} and {{nowrap|''R''<sup>''n''×''n''</sup>}}{{sfnp|Artin|2018|loc=Example 3.3.6(a)|ps=}}).  Some sets of infinite matrices form '''infinite matrix rings'''.  A subring of a matrix ring is again a matrix ring.  Over a [[rng (algebra)|rng]], one can form matrix rngs.

When ''R'' is a commutative ring, the matrix ring M<sub>''n''</sub>(''R'') is an [[associative algebra]] over ''R'', and may be called a '''matrix algebra'''.  In this setting, if ''M'' is a matrix and ''r'' is in ''R'', then the matrix ''rM'' is the matrix ''M'' with each of its entries multiplied by ''r''.