Ring of Square Matrices over Ring with Unity

Theorem

Let $R$ be a ring with unity.

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

Let $\struct {\map {\MM_R} n, +, \times}$ denote the ring of square matrices of order $n$ over $R$.

Then $\struct {\map {\MM_R} n, +, \times}$ is a ring with unity.

Proof

From Ring of Square Matrices over Ring is Ring we have that $\struct {\map {\MM_R} n, +, \times}$ is a ring.

As $R$ has a unity, the unit matrix can be formed.

The unity of $\struct {\map {\MM_R} n, +, \times}$ is this unit matrix.

$\blacksquare$

Notation

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

$\struct {\map {\MM_R} n, +, \times}$

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.

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields: Example $3$