Matrix ring - Knowlpedia

Let $R$ be a ring and $n\ge 1$ . The matrix ring $M_n(R)$ consists of the $n\times n$ matrices over $R$ , with entrywise addition and multiplication

(AB)_{ij}=\sum_{k=1}^n A_{ik}B_{kj}.

Remarks

If $R$ is unital, then $M_n(R)$ is unital with identity matrix $I_n$ . Even when $R$ is commutative, $M_n(R)$ is generally noncommutative for $n>1$ . If $D$ is a division ring, then $M_n(D)$ is simple, and its center consists of scalar matrices with entries in $Z(D)$ .

Examples

$M_2(\mathbb Z)$ is a noncommutative ring with identity.
For a prime $p$ , the ring $M_3(\mathbb F_p)$ has $p^9$ elements.
The ring $M_1(R)$ is canonically isomorphic to $R$ .