Existence of a non-semisimple ring such that every module over it has a simple submodule

Question

Does there exists a ring $R$ which is not semisimple but every module over it has a simple submodule?

Answer 1

Here is an example: $R = \mathbb{Z}/4$.

Take a non-zero $R$-module $M$ which is simply an abelian group annihilated by $4$.

If $2M$ is non-zero, it will be a non-zero $R/2R \simeq \mathbb{Z}/2$ module, but $\mathbb{Z}/2$ is a field, done.

If $2M$ is $0$ then $M$ is a $R/2R$ module, ... , again done.

Obs: Works similarly for $\mathbb{Z}/p^2$, for $\mathbb{Z}/p^n$ and, more generally, for every Artinian ring (http://en.wikipedia.org/wiki/Artinian_ring)