Ring of Square Matrices over Field is Ring with Unity

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Theorem

Let $F$ be a field.

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

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


Then $\struct {\map {\mathcal M_F} n, +, \times}$ is a ring with unity, but is not a commutative ring.


Proof

We have by definition that a field is a division ring which is also commutative.

Hence $F$ is a commutative ring with unity.

So, from Ring of Square Matrices over Commutative Ring with Unity we have that $\struct {\map {\mathcal M_F} n, +, \times}$ is a ring with unity.

From [Matrix Multiplication on Square Matrices over Ring with Unity is not Commutative‎]], we have that $\struct {\map {\mathcal M_F} n, +, \times}$ is not a commutative ring.


Hence the result.

$\blacksquare$


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