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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=â_iâIM_i where I is an index set and for each i, MinAss(M_i)={P_i},R/P_i is a principal ideal domain and M_i=S_iâââT_i with semisimple and torsionfree (R/P_i)-modules S_i and T_i, respectively. As an application of our âstructure theorem,â we give a characterization of commutative rings for which each proper ideal is virtually semisimple.
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