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Paper   IPM / M / 16041
School of Mathematics
  Title:   Structure of virtually semisimple modules over commutative rings
  Author(s):  Mahmood Behboodi (Joint with A. Daneshvar and M. R. Vedadi)
  Status:   Published
  Journal: Comm. Algebra
  Year:  2020
  Pages:   DOI: 10.1080/00927872.2020.1723611
  Supported by:  IPM
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∈IM_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_i ⊕ T_i with semisimple and torsionfree (R/P_i)-modules S_i and T_i, respectively. As an application of our “structure theorem,” we give a characterization of commutative rings for which each proper ideal is virtually semisimple.

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