Projective cover of a simple module and indecomposable factor modules

Question

Let $P$ a non-zero projective module. Prove that $P$ is the projective cover of a simple module if and only if every non-zero factor module of $P$ is indecomposable.

Thank you.

Answer 1

(Assuming you meant, as is likely, that $P$ is a projective module)

Suppose $P$ is a projective cover of a simple module. This is clearly equivalent to saying that $P$ has a maximal superfluous submodule $K$. We claim that every proper submodule of $P$ is superfluous. If $J$ is a proper submodule, then either $J+K=P$ or $J\subseteq K$. The former is impossible since $K$ is superfluous and $J$ is proper. Thus all proper submodules are contained in the superfluous submodule $K$, and hence they are all superfluous.

It is then easy to establish that, for any superfluous submodule $J$, $P/J$ is indecomposable, and since all proper submodules are superfluous, all nonzero quotients are indecomposable.

Conversely, suppose every nonzero quotient is indecomposable. Begin with the fact that every nonzero projective module has a maximal submodule, call it $K$. Deduce that $K$ is superfluous by contradiction. If it weren't, there would be a proper submodule $J$ such that $K+J=P$. But this would imply that $P/(K\cap J)$ is decomposable, contrary to assumption.

Since $K$ is maximal and superflous, $P/K$ is a simple module covered by $P$.