I have seen in the paper "projective injective modules" by J. P. Jans that "every module over a semisimple ring with minimum condition is both projective and injective, while over the integers only the zero module has this property."
I think that every module over a semisimple algebra is semisimple, so every module over a semisimple ring with minimum condition is also semisimple and both projective and injective , is that right? Another question is that how to get only the zero module over the integers is projective injective?(My work: Since $\mathbb{Z}=1\mathbb{Z}$, the projective modules of $\mathbb{Z}$ are $0$ and $\mathbb{Z}$. But I can't prove $\mathbb{Z}$ is not an injective $\mathbb{Z}$-module.)
Yes, over a semisimple ring every module is projective and every module is injective, because every exact sequence splits.
Over the integers, a module (that is, an abelian group) $E$ is injective if and only if it is divisible: for each $x\in E$ and each $n>0$, there exists $y\in E$ with $ny=x$.
On the other hand, any projective module is a submodule of a free module (free abelian group) and free abelian groups have no nonzero divisible subgroup.