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

t. All rings are commutative. Let M be a module. We introduce the property (P): Every endomorphism of M has a non-trivial invariant submodule. We determine the structure of all vector spaces having (P) over any field and all semisimple modules satisfying (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 (P) as that of the rings R for which R/m is an algebraically closed field for every maximal ideal m of R.