I need an example of a non-commutative ring containing 4 elements. I considered the following example of $M_2(Z_2)$ containing the following four elements:
$\begin{bmatrix}0&0\\0&0\end{bmatrix} \begin{bmatrix}1&0\\0&0\end{bmatrix} \begin{bmatrix}0&1\\0&0\end{bmatrix} \begin{bmatrix}1&1\\0&0\end{bmatrix}$
Here clearly the set is non-commutative. I have also checked that the set forms a ring. But I want to be sure whether the set forms a ring or not.
Can anyone tell me whether it is forming a ring or not?
EDITS
Previously the elements I chose were not closed under addition. So I replaced the identity element with another element and checked that now it is closed under addition as well as non commutative.
Is it correct now?