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.