Strongly T-Semisimple Modules And Strongly T-Semisimple Rings

We define and investigate strongly t-semisimple modules as a special class of t-semisimple modules. A module M is called strongly t-semisimple if M is a t-semisimple module and every nonsingular cyclic submodule mR (0≠ m ∈ M) is simple. The class of strongly t-semisimple is closed under submodules, homomorphic image, but not closed under direct sum. Many equivalent conditions for a module M to be strongly t-semisimple are found. We also define and investigate right strongly t-semisimple rings as a special class of t-semisimple rings. A ring R is called right strongly t-semisimple ring if RR is strongly t-semisimple module. Various characterizations of strongly t-semisimple ring are given. In the end, we discuss the connection between the strongly t-semisimple ring and the other rings.In chapter 1 of this thesis, we will mainly put forward the background and significance of this paper, and offer several closely related concepts and theorems in the article also, such as Z%-torsion modules, t-semisimple modules, t-essential modules etc.In chapter 2 of this thesis, we introduce the notion of strongly t-semisimple modules. Some examples are given to strongly t-semisimple modules. Submodules and homomorphic image of strongly t-semisimple modules inherit the property, but not direct sum of strongly t-semisimple modules. We will show that strongly t-semisimple modules admit many other characterizations. The main result is:Theorem 2.4 The following statements are equivalent for a module M:(1) M is strongly t-semisimple;(2) M/Z2{M) is very semisimple;(3) M=Z2(M)(?- M’, where M’ is a (nonsingular) very semisimple module;(4) Every nonsingular submodule of M is a direct summand and every nonsingular cyclic submodule mR (0≠ m∈M) is simple;(5) Every submodule of M which containsZ2(M) is a direct summand. (m1+m2) R is simple whenever m1R and m2R are simple where ma and m2 are in M and m1+m2≠ 0.Furthermore, we have the following conclusion:Corollary 2.7 The following statements are equivalent for a module M:(1) M is strongly t-semisimple;(2) Every nonsingular cyclic submodule of M is simple and is a direct summand;(3) Every nonsingular cyclic submodule of M is simple and every nonsingular finitely gener-ated submodule of M is a direct summand;Corollary 2.8 A module M is strongly t-semisimple if and only if Rad M is Z2-torsion and every nonsingular cyclic submodule of M is simple and has a weak supplement.Proposition 2.9 The following statements are equivalent for a module M:(1) M/Rad M is strongly t-semisimple;(2) M=M1(?) M1, such that M1 is very semisimple and Rad M1 tes M2.Corollary 2.10 M is strongly t-semisimple if and only if M/Rad M is strongly t-semisimple and Rad M is Z2-torsion.In chapter 3 of this thesis, we deal with strongly t-semisimple rings. Some examples are given to strongly t-semisimple rings. We will give an example of a right t-semisimple rings which is not strongly t-semisimple rings. Several characterizations for strongly t-semisimple rings are obtained in theorem 3.2, theorem 3.4 and theorem 3.6.Theorem 3.2 The following statements are equivalent for a ring R:(1) R is right strongly t-semisimple;(2) Every cyclic submodule R-module is strongly t-semisimple;(3) Every nonsingular cyclic submodule.R-module is simple;(4) For every cyclic R-module M, there is a simple submodule M’such that M= Z2 (M)(?)M;:(5) Every cyclic projective R-module is strongly t-semisimple.Theorem 3.4 The following statements are equivalent for a ring R:(1) R is strongly t-semisimple;(2) Every free t-module is strongly t-semisimple;(3) Every t-module is strongly t-semisimple;(4) Every nonsingular t-module is very semisimple;(5) Every projective.R-module is strongly t-semisimple.Theorem 3.6 The following statements are equivalent for a ring R:(1) R is right strongly t-semisimple;(2) Z2{Rr) is a maximal right ideal of R;(3) R is a direct product of two rings, one is a right Z2-torsion ring and the other is a division ring.