Structure of virtually semisimple modules over commutative rings

2020/02/17 by M. Behboodi, Mahmood Behboodi, A. Daneshvar +2
Mathematics · #Rings, Modules, and Algebras #Algebraic structures and combinatorial models #Advanced Topics in Algebra

paper · doi:10.1080/00927872.2020.1723611

Abstract

Modules in which every submodule is isomorphic to a direct summand is called virtually semisimple. In this article, we carry out a study of virtually semisimple modules over a commutative ring R. A structure theorem of finitely generated virtually semisimple R-modules is given. Also, it is proven that if every submodule of an R-module M is virtually semisimple, then M=⊕ i∈IMi where I is an index set and for each i, MinAss(Mi)=Pi,R/Pi is a principal ideal domain and Mi=Si ⊕ Ti with semisimple and torsionfree (R/Pi)-modules Si and Ti, respectively. As an application of our “structure theorem,” we give a characterization of commutative rings for which each proper ideal is virtually semisimple.

Citations

• Modules with chain conditions up to isomorphism
• Artinian dimension and isoradical of modules

Related

• Virtually semisimple modules and a generalization of the Wedderburn-Artin theorem
• Virtually Semisimple Modules and a Generalization of the\n Wedderburn-Artin Theorem
• Virtually regular modules
• Two Generalizations of the Wedderburn-Artin Theorem with Applications
• T-Semisimple Modules and T-Semisimple Rings