2013/05/09 by Sh. Asgari, A. Haghany, Y. Tolooei
Mathematics · Chemistry · #Rings, Modules, and Algebras #Algebraic structures and combinatorial models #Oxidative Organic Chemistry Reactions
paper · doi:10.1080/00927872.2011.653065
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. Key Words: Nonsingular and Z 2-torsion modulesT-essential submodulesT-semisimple modules2010 Mathematics Subject Classification: 16D1016D7016D9016P70 ACKNOWLEDGMENT The authors wish to express their deep gratitude to the referee for many valuable comments which helped to improve the present version of this article. The research of the first author was in part supported by a grant from IPM (No. 90130036). Notes Communicated by T. Albu.