Let $R$ be a commutative artinian ring with identity. It is true that for $n>0$ the matrix ring $M_n(R)$ is left and right artinian?
Yes. This is because the Artinian condition is a Morita invariant condition. This means that $R$ and any ring Morita equivalent to it (including its matrix rings) is left and right Artinian.
You can prove it without the full brunt of Morita theory, though.
The way to connect (generalized by Morita) $R$ and $M_n(R)$ is to examine $R^n$ as an $M_n(R)$ module with the obvious (matrix multiplication) module action.
If you can get ahold of it at a library, or take a look in googlebooks, Chapter 7 of Lam's Lectures on Modules and Rings has a very accessible introduction to seeing this connection between matrix rings and their base rings