# Modules with maximal submodules and projective dimension

If $R$ is a left noetherian ring, then every finitely generated left $R$-module $M$ is noetherian, and hence every proper submodule of $M$ is contained in some maximal submodule of $M$.

Is it possible to weaken the hyphotesis? I.e. when can we assure that a left (non-zero) $R$-module has a maximal submodule?

Suppose now that $R$ is left artinian ring with identity. I was wondering, for instance, if it was true that (non-zero) left $R$-modules of finite projective dimension have some maximal submodule. If this latter statement is not true, would it be true if the base ring was an Artin algebra?

I am asking this to try to solve the problem I posted in "Artinian rings with zero finitistic dimension" in full generality.

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When H. Bass was investigating perfect rings, he found that for a right perfect ring $R$, every nonzero right $R$ module has a maximal submodule, and that $R$ does not contain infinite sets of orthogonal idempotents. He conjectured that the converse might also be true.