Matrix ring

In abstract algebra, a matrix ring is any collection of matrices over some ring R that form a ring under matrix addition and matrix multiplication (Lam 1999). The set of n × n matrices with entries from R is a matrix ring denoted M_n(R), as well as some subsets of infinite matrices which form infinite matrix rings. Any subring of a matrix ring is a matrix ring.

When R is a commutative ring, the matrix ring M_n(R) is an associative algebra, and may be called a matrix algebra. For this case, if M is a matrix and r is in R, then the matrix Mr is the matrix M with each of its entries multiplied by r.

This article assumes that R is an associative ring with a unit 1 ≠ 0, although matrix rings can be formed over rings without unity.

and ${\displaystyle {\begin{bmatrix}1&1\\0&0\end{bmatrix}}{\begin{bmatrix}1&0\\0&0\end{bmatrix}}={\begin{bmatrix}1&0\\0&0\end{bmatrix}}\,}$. This example is easily generalized to n×n matrices.

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