If $R$ is a unital ring and $M_{2\times 2}(R)$ is a commutative ring, then $R$ is a trivial ring because if $\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}=\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}\begin{pmatrix}0 & 1 \\ 0 & 1 \end{pmatrix}=\begin{pmatrix}0 & 1 \\ 0 & 1 \end{pmatrix}\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}=\begin{pmatrix}0 & 0\\0 & 0\end{pmatrix}, $ then $0=1.$
However, $M_{2\times 2}(R)$ can be commutative for possibly non-unital rings of any size. That is, for any cardinal number $\kappa$ (finite or not), there exists an abelian group of order $\kappa.$ Equipping this group with the zero multiplication gives a rng $R$ such that $M_{2\times 2}(R)$ has zero multiplication and so is commutative. The trivial ring is also in this class.
Are there any examples of rngs $R$ whose multiplication is non-zero and such that $M_{2\times 2}(R)$ is commutative?