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

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


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

Similarly, it is clear that $\struct {\map {\MM_R} n, \times}$ is a semigroup, as Matrix Multiplication over Order n Square Matrices 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:

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

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.