Let N be a submodule of the module M. Suppose M/N and N are semi-simple. Does it follow that M is semi simple?

Question

Let N be a submodule of the module M. Suppose M/N and N are semi-simple. Does it follow that M is semi simple?

I think the answer is yes but I am not sure how to prove it. Any help would be appreciated.

Answer 1

No. Consider the ring $R=\mathbb C[t]/(t^2)$. The regular module $M=R$ contains the submodule $N=(t)$ which is simple, and $M/N$ is also simple, yet $M$ is not semisimple.

In fact, under sensible hypotheses if a ring is such that your statement does hold, then the ring must be semisimple. For example, if the ring is a finite dimensional algebra over a ring (or even an artin algebra)

Answer 2

Adding on Qiaochu's comment and trying to come up with an easier, perhaps, counter-example than that given by Mariano: the only simple $\mathbb{Z}-\,$modules (which are the same as abelian groups) are the cyclic groups of prime order, thus: take Klein's viergrup $$\,M:=C_2\times C_2\,\,,\,\,N:= \{1\}\times C_2\,\,,\,\,M/N\cong C_2$$ Both $M\,,\,M/N\,$ are even simple, as mentioned by Qiaochu