Symmetry And Reduction Of Rings And Modes

This dissertation, consisting of six chapters, is concerned with the reduced and symmetric properties of modules and rings.Chapter 1 summarizes the background and main results obtained in this thesis.Chapter 2 briefly introduces some elementary concepts and a few notational oddities that will be used through the thesis.Chapter 3 discusses the ideal symmetric modules. As a natural gener-alization of the concepts of symmetric modules and ideal symmetric rings,we first introduce the concept of ideal symmetric modules,and give some equivalent charaterizations of the concept. Second,we discuss the modules of fractions, let S be a subset of some regular elements of a ring R satisfying the right Ore conditions,and M a right R-module,it is proved that the module of fractions M[S-1] is an ideal symmetric R[S-1]-module if M is ideal symmetric.Chapter 4 is concerned with the reduced property of modules, and some equivalent conditions of reduced modules are obtained by using semiprime and symmetric as well as zero-insertive properties of modules.Let M be a R-module.For example, it is proved that if M is regular, then M is reduced if and only if M is symmetric, if and only if M is zero-insertive,and if and only if it is abelian.If M is symmetric then M/Z(M) is reduced,where Z(R) is the singular submodule of M.It is also shown that a flat module over a classical com-pletely semiprime ring (respectively, zero-insertive ring) is classical completely semiprime (respectively,zero-insertive).Chapter 5 is on the J-reversible rings,?-reversible rings, and n-reversible rings, as well as central reversible rings.In this chapter,the basic properties of these rings are discussed, and some equivalent charaterizations of J-reversible rings are obtained by using the upper triangular matrix rings or generalized matrix rings.Let R be a ring, it is proved that R is reduced if it satisfies one of the following conditions:(1) R is a ?-reversible ring such that each singular simple left R-module is nil-injective; (2) R is a central reversible ring such that every singular simple left R-module is nil-injective; (3) R is a ?-reversible and SF-ring; (4) R is a J-reversible ring with every nilpotent element regular. It is also shown that a J-reversible ring R is a clean ring if and only if it is an exchange ring, and in this case, R has the stable range 1.Chapter 6 is the peroration of this dissertation.