I want to figure out whether $R$ is isomorphic to $S$, where $R = \mathbb{R}[G]$, where $G = \mathbb{Z}/2 \times \mathbb{Z}/2$, and $S = M_4(\mathbb{R})$.
It seems that they might not be isomorphic, since the obvious isomorphism $\phi \left( \begin{matrix} a & b \\ c & d \end{matrix}\right) = a(0,0)+b(0,1)+c(1,0)+d(1,1)$ doesn't work because multiplication is not preserved, I think. Am I doing something wrong or are these actually not isomorphic?
A group ring with a commutative ring and an abelian group is obviously a commutative ring. $M_2(\mathbb R)$ is obviously not a commutative ring.
Since a group ring of a nontrivial group is never simple (the augmentation ideal is a nontrivial ideal) a group ring over such a group can never be a matrix ring over a field (which is always simple.)