Ring of Square Matrices over Ring is Ring

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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.


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.



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.