Which modules have exactly one maximal submodule?

Question

Let $(R, {\frak m})$ be a Noetherian local ring. When a finitely generated module has exactly one maximal submodule?

Answer 1

If $M \neq 0$ has exactly one maximal submodule, the same holds for $M/\mathfrak mM$. This is a $R/\mathfrak m$-vector space. Clearly a vector space with exactly one maximal subspace must be one-dimensional. By Nakayama, $M$ is cyclic, i.e. isomorphic to $R/\operatorname{Ann}(M)$.