A semisimple ring $R$ with $1$ (but not necessarily a commutative one) considered as left-$R$ module is a direct sum
$$R\cong L_1\oplus L_2 \oplus \cdots\oplus L_n$$ such that for some $e_i$ are in $R$
- $L_i=Re_i$ simple left ideal of $R$
- $e_ie_j=0$ for different $i,j$
- $e^2_j=e_j$
- $\sum e_i=1$ the unit of the ring
Now I want to show $R_1,R_2$ are semisimple rings with $1$ if and only if $R_1\times R_2$ is semisimple.
My problem is how to see their structure, if $R_1,R_2$ semisimple how to construct and see the decomposition of $R_1\times R_2\cong L_1\oplus L_2 \oplus \cdots\oplus L_n\times K_1\oplus K_2 \oplus \cdots\oplus K_m$
$L$s for $R_1$, $K$s for $R_2$ but these summands have $R_1$ and $R_2$ structre.
By the way how to solve this problem in more efficient and clever way, can we use homological algebra, another decompositions?
That's just fine, because you can define an $R$ structure on $L_i$ as $(r_1,r_2)x=r_1x$ and an $R$ structure on $K_i$ as $(r_1,r_2)x=r_2x$ and it turns out that with this action they are all simple as $R$ modules too.
I think this is about as efficient as it gets. If by clever you mean "fancy" or "sophisticated" then I think that is almost always possible.