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Math 2.0

Linear and multilinear algebra; matrix theory Object

Matrix ring

Description

Let $R$ be a ring and let $n \ge 1$ be an integer. The matrix ring $M_n(R)$ is the set of $n \times n$ matrices over $R$ , with matrix addition and matrix multiplication. It is a ring, with the zero matrix as zero and the identity matrix $I_n$ as unit. If $R$ is commutative, then $M_n(R)$ is also an algebra over $R$ , with scalar multiplication.

Remarks

For $n \ge 2$ and $R \ne 0$ , the ring $M_n(R)$ is not commutative and has nonzero elements whose product is zero. The units of $M_n(R)$ are the invertible matrices, and they form the general linear group. For a field $F$ , the ring $M_n(F)$ is isomorphic to the ring of endomorphisms of $F^n$ .

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Further reading

Web https://en.wikipedia.org/wiki/Matrix_ring, lead section

About this page

Tag	`0FHVPRAE60`
URI	`m2://0FHVPRAE60`
Kind	object
Writing	W1 stated
MSC	15A99, 16S50
Instance of	Matrix
Same as	Wikidata Q2915729 Wikipedia Matrix ring

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section Invariants: a table of the main invariants (0 of 1)

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