Ring of Square Matrices over Real Numbers

Theorem

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, but is not a commutative ring.

Proof

Recall that Real Numbers form Field.

The result follows directly from Ring of Square Matrices over Field is Ring with Unity.

$\blacksquare$

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.

Examples

$2 \times 2$ Real Matrices

Let $\struct {\map {\MM_\R} 2, +, \times}$ denote the ring of square matrices of order $2$ over the real numbers $\R$.

Then $\struct {\map {\MM_\R} 2, +, \times}$ forms a ring with unity which is specifically not commutative and also not an integral domain.

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 54$. The definition of a ring and its elementary consequences