# Definition:Ring of Square Matrices

## Definition

Let $R$ be a ring.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $\map {\mathcal M_R} n$ denote the $n \times n$ matrix space over $R$.

Let $+$ denote the operation of matrix entrywise addition.

Let $\times$ be (temporarily) used to denote the operation of conventional matrix multiplication.

The algebraic structure:

- $\struct {\map {\mathcal M_R} n, +, \times}$

is known as the **ring of square matrices of order $n$ over $R$**

## Notation

When referring to the operation of **matrix multiplication** in the context of the ring of square matrices:

- $\struct {\map {\mathcal M_R} n, +, \times}$

we *must* have some symbol to represent it, and $\times$ does as well as any.

However, we do *not* use $\mathbf A \times \mathbf B$ for matrix multiplication $\mathbf A \mathbf B$, as it is understood to mean the vector cross product, which is something completely different.

## Also see

- Ring of Square Matrices over Ring is Ring
- Ring of Square Matrices over Ring with Unity
- Ring of Square Matrices over Field is Ring with Unity

- Results about
**rings of square matrices**can be found here.