Ring of Square Matrices over Real Numbers/Examples/2 x 2

Example of Ring of Square Matrices over Real Numbers
Let $\struct {\map {\mathcal M_\R} 2, +, \times}$ denote the ring of square matrices of order $2$ over the real numbers $\R$.

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

Proof
We check the ring axioms:

Let $\mathbf A = \sqbrk a_{i j}$, $\mathbf B = \sqbrk b_{i j}$ and $\mathbf C = \sqbrk c_{i j}$ where:
 * $i, j \in \set {1, 2}$
 * $a_{i j}, b_{i j}, c_{i j} \in \R$

be arbitrary real elements of $\map {\mathcal M_\R} 2$.

$A0$: Closure under Addition
From Matrix Entrywise Addition: $2 \times 2$ Real Matrices, matrix addition on $\map {\mathcal M_\R} 2$ is closed.

$A1$: Associativity of Addition
Thus matrix addition is associative on $\map {\mathcal M_\R} 2$

$A2$: Commutativity of Addition
Thus matrix addition is commutative on $\map {\mathcal M_\R} 2$.