T-Semisimple Modules and T-Semisimple Rings |
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Authors: | Sh. Asgari A. Haghany Y. Tolooei |
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Affiliation: | 1. Department of Mathematical Sciences , Isfahan University of Technology, Isfahan, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM) , Tehran , Iran sh_asgari@math.iut.ac.ir;3. Department of Mathematical Sciences , Isfahan University of Technology , Isfahan , Iran |
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Abstract: | We define and investigate t-semisimple modules as a generalization of semisimple modules. A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t-essential in N. T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules. Many equivalent conditions for a module M to be t-semisimple are found. Accordingly, M is t-semisiple, if and only if, M = Z 2(M) ⊕ S(M) (where Z 2(M) is the Goldie torsion submodule and S(M) is the sum of nonsingular simple submodules). A ring R is called right t-semisimple if R R is t-semisimple. Various characterizations of right t-semisimple rings are given. For some types of rings, conditions equivalent to being t-semisimple are found, and this property is investigated in terms of chain conditions. |
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Keywords: | Nonsingular and Z 2-torsion modules T-essential submodules T-semisimple modules |
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