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

Authors : Mohamed BENSLIMANE, Hanane EL CUERA, Rachid TRIBAK

Pages : 107-119

Doi:10.24330/ieja.852029

View : 23 | Download : 13

Publication Date : 2021-01-05

Article Type : Research Paper

Abstract :All rings are commutative. Let $M$ be a module. We introduce the property $insert ignore into journalissuearticles values({\bf P});$: Every endomorphism of $M$ has a non-trivial invariant submodule. We determine the structure of all vector spaces having $insert ignore into journalissuearticles values({\bf P});$ over any field and all semisimple modules satisfying $insert ignore into journalissuearticles values({\bf P});$ 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 $insert ignore into journalissuearticles values({\bf P});$ as that of the rings $R$ for which $R/\mathfrak{m}$ is an algebraically closed field for every maximal ideal $\mathfrak{m}$ of $R$.
Keywords : Algebraically closed field, characteristic polynomial of a matrix an endomorphism, fully invariant submodule, homogeneous semisimple module, invariant submodule