Ring of Square Matrices over Ring is Ring

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Theorem

Let $R$ be a ring.

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

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


Then $\struct {\map {\mathcal M_R} n, +, \times}$ is a ring.


Proof

From Matrix Entrywise Addition over Group forms Group we have that $\struct {\map {\mathcal M_R} n, +}$ is an abelian group, because $\struct {R, +}$ is itself an abelian group.



Similarly, it is clear that $\struct {\map {\mathcal M_R} n, \times}$ is a semigroup, as Matrix Multiplication is Closed and Matrix Multiplication is Associative.

Finally, we note that Matrix Multiplication Distributes over Matrix Addition.

$\blacksquare$


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.


Sources