Let $\mathbb{F_2}: = \mathbb{Z}/2\mathbb{Z}$ be the field with two elements and consider the ring $R$ = $M(2,\mathbb{F_2})$ of 2 x 2- matrices with coefficients in $\mathbb{F_2}$.
Let $A = \begin{bmatrix} 1&0\\1&0 \end{bmatrix}$ and $B = \begin{bmatrix} 0&1\\0&1 \end{bmatrix}$ be two matrices in $R$. Let $O$ bet the zero matrix.
Let $S=\{O, A, B, A + B\}\subset R$.
If we write $+s$ and $*s$ for the induced maps
$+s$ : $S$ $*$ $S$ $\rightarrow$ $R$
$*s$ : $S$ $*$ $S$ $\rightarrow$ $R$
I have to show that $*s$ is contained in $S$ and using that I have to show that ($S, +s, *s$) is a non- unitary, non- commutative ring of order 4.
Just try to find out $A^2$, $B^2$, $AB$ & $BA$ and use $2=0$.