I recently read the concept of semi-simplicity of a (not necessarily commutative) ring.
A ring $R$ is said to be semi-simple if $R$ as a left module over itself is a semi-simple module (This in turn means that each submodule (i.e, left ideal) of $R$ is a direct summand for $R$.)
I am grappling with the following question:
If $R$ is semi-simple, then so is the matrix ring $M_n(R)$.
It is a theorem that each left module of a semi-simple ring is a semi-simple module. Thus $M_n(R)$ is a semi-simple module of $R$. But this doesn't help me.
I need to show that each left ideal of $M_n(R)$ is a direct summand for $M_n(R)$. Is there a characterization of the left ideals of $M_n(R)$ in terms of the left ideals of $R$?
Lastly, does the converse also hold?
If $M_n(R)$ is a semi-simple ring, then is $R$ necessarily a semi-simple ring?
Thanks.
Show that for any simple left ideal $S \subseteq R$ and index $j$ $$R_{S, j} = \left\{A \in M_n(R) \ \middle| \ A_{ij} \in S \ \text{and} \ A_{ik} = 0 \ \forall i, k \neq j\right\}$$ is a simple left ideal in $M_n(R)$ and as a module over itself $M_n(R)$ is the direct sum of these when $S$ varies over all simples and $j$ varies over all indices.