If $R/I$ and $R/J$ are semisimple, then so is $R/I\cap J$. 
Let $R$ be a not necessarily commutative ring. If $I$ and $J$ are (two-sided) ideals in $R$ such that $R/I$ and $R/J$ are both semi-simple rings, then so is $R/I\cap J$.

I tried the following:
Define $f:R\to R/I\times R/J$ as $r\mapsto (r+I, r+J)$. Then $\ker f=R/I\cap J$.
Thus the ring $R/I\cap J$ is isomorphic to $f(R)$, the latter being a subring of $R/I\times R/J$.
Does anybody see if we can show that $f(R)$ is a semi-simple ring?
Is there some other way to do this?
Thanks.
 A: By definition, $R$ is a semisimple ring iff it is a semisimple left module over itself.
Let $K$ be a two-sided ideal. The left $R$ modules of the form $R/K$ have some extra structure on them given by the fact that scalar multiplication is intimately tied up with multiplication in the ring. This especially helps us when we want to multiply by $1_R$, for instance: taking $K=0$ we can show that if $R$ is a semisimple left module over itself then it is in fact a direct sum of finitely many simple ideals. 
When we use the normal scalar multiplication as above, $R/K$ as an $R$ module has essentially all of the same properties as it does as an $R/K$ module; all we've done is gotten rid of the extra noise and made the action faithful by modding out the ring of scalars by the annihilator $\text{Ann}_R(R/K)=K$. In particular, the left $R/K$ submodules of $R/K$ are the same as the left $R$ submodules of $R/K$, and a submodule is simple in one case iff it is simple in the other. So $R/K$ is a semisimple $R$ module iff it is a semisimple $R/K$ module. 
We can then use the assumption of the problem to conclude that $R/I$ and $R/J$ are in fact semisimple $R$ modules. $R/(I \cap J)$ is isomorphic to an $R$ submodule of $R/I \oplus R/J$, by the map you mention, and note that the scalar multiplication on $R/(I \cap J)$ is again the multiplication we'd expect. Direct sums of semisimple modules are semisimple, so $R/I \oplus R/J$ is semisimple, and submodules of a semisimple module are semisimple, so $R/(I \cap J)$ is a semisimple $R$ module.
Now again we note that $R/(I \cap J)$ is a semisimple $R$ module iff it is a semisimple $R/(I \cap J)$ module.
