MODULES FOR WHICH EVERY ENDOMORPHISM HAS A NON-TRIVIAL INVARIANT SUBMODULE

Abstract

All rings are commutative. Let $M$ be a module. We introduce the property $({\bf P})$: Every endomorphism of $M$ has a non-trivial invariant submodule. We determine the structure of all vector spaces having $({\bf P})$ over any field and all semisimple modules satisfying $({\bf P})$ over any ring. Also, we provide a structure theorem for abelian groups having this property. We conclude the paper by characterizing the class of rings for which every module satisfies $({\bf P})$ as that of the rings $R$ for which $R/\mathfrak{m}$ is an algebraically closed field for every maximal ideal $\mathfrak{m}$ of $R$.