Matrix ring

Let $R$ be a ring and let $n\ge 1$. The matrix ring $M_n(R)$ is the set of all $n\times n$ matrices with entries in $R$, with addition defined entrywise and multiplication defined by the usual row-by-column rule.

If $R$ is unital , then $M_n(R)$ is unital with identity matrix $I_n$. For many structural properties, matrix rings behave like “noncommutative polynomial extensions”: for instance, $M_n(D)$ over a division ring $D$ is simple and has center given by scalar matrices from $Z(D)$.

Examples:

• $M_2(\mathbb{Z})$ is a noncommutative ring with identity $I_2$.
• For a prime $p$, $M_3(\mathbb{F}_p)$ is a finite ring of size $p^9$.
• $M_1(R)$ is canonically isomorphic to $R$.