2016/03/17 by Mahmood Behboodi, Asghar Daneshvar, Behboodi, Mahmood +3
Mathematics · #13F10 #16D70 #16S50 #FOS: Mathematics #Primary 16D60 #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1603.05647
openalex publication_date 2016/03/17 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
By any measure, semisimple modules form one of the most important classes of\nmodules and play a distinguished role in the module theory and its\napplications. One of the most fundamental results in this area is the\nWedderburn-Artin theorem. In this paper, we establish natural generalizations\nof semisimple modules and give a generalization of the Wedderburn-Artin\ntheorem. We study modules in which every submodule is isomorphic to a direct\nsummand and name them it virtually semisimple modules. A module RM is\ncalled it completely virtually semisimple if each submodules of M is a\nvirtually semisimple module. A ring R is then called it left ( it\ncompletely) it virtually semisimple if RR is a left (compleatly)\nvirtually semisimple R-module. Among other things, we give several\ncharacterizations of left (completely) virtually semisimple rings. For\ninstance, it is shown that a ring R is left completely virtually semisimple\nif and only if R \≅ \∏ i=1^ k Mni(Di) where k, n1, ...,nk\∈\n\ℕ and each Di is a principal left ideal domain. Moreover, the\nintegers k,~ n1, ...,nk and the principal left ideal domains D1, ...,Dk\nare uniquely determined (up to isomorphism) by R.\n