Strong Regularity, The Study Of Several Class Ring

It is known that von Neumann regular ring must be SF ring, but the converse is still a open question. The discussion on the regularity of SF ring is an important question of ring theory. The famous result given by professor Rege (1986) is that the reduced SF ring is a strongly regular nng, then it is a von Neumann regular ring. ZI ring is an important conception of ring theory as a generalization of reduce ring. It give a great effort on the study of strongly regular ring. As a generalization of ZI ring, main aim of this paper is to discuss the regularity of WZI ring,PZI ring and SF ring. On the basis of WZI ring, we give a new ring NZI ring and have the preliminarily study, get some significance results.In the whole, this paper has five chapters. The first chapter, we give the study background and some preliminaries needed in the paper. In the second chapter, we study the some basic propositions of WZI ring PZI ring and NZI ring. The direct sum, direct product and the finite subdirect product is closed. We proof that R[x] is WZI ring if and only if R[x,x-1] is also WZI ring. In the third chapter, we study some application of WZI ring. We proof that if R is a left WZI ring and every simple single left R-module is YJ-injective, then R is a reduced weak regular ring. So we improve the conclusion that if R is a ZI ring and every simple single left R-module is YJ-injective, then R is a reduced weak regular ring proposed by Kim, Nam, and Kim (2003). It is known that R is a strongly regular ring if and only if R is an Abelian ring and a von Neumann ring. In the paper, we proof that R is a strongly regular ring if and only if R is a left WZI ring and a von Neumann ring if and only if R is a left PZI ring and a von Neumann ring. In the fourth chapter, we study the regularity of the SF ring R in the condition of R is a left and right WZI ring and PZI ring. We proof that (1) R is a left WZI ring, then R/Z1(R)is a ZI ring.(2) R is a left PZI ring, then r/Zr(R)is a ZI ring. By the result, we have (3) R is a strongly regular ring if and only if R is a left WZI ring and a left SF ring if and only if R is a right PZI ring and a left SF ring.(4) We proof that R is a strongly regular ring if and only if R is a right WZI ring and a left SF ring if and only if R a is left PZI ring and a left SF ring.Those results all improve if R is a reduced left SF ring, then R is a strongly regular ring proposed by Rege (1999). In the last chapter, by the generalization of WZI ring, we give a new ring NZI ring. In the chapter, we proof that (1) If R is a NZI ring and a left MC2ring and every simple singular left R-module is nil-injective, then R is a biregular ring.(2) We proof that if R is a semiprime ring, R is a reduced ring if and only if R is a NZI ring if and only if R is a left WZI ring if and only if R is a left PZI ring. By the result, we improve if R is a semiprime ring, R is a reduced ring if and only if R is a ZI ring given by Kim and Lee (2003). In the paper, we proof that if R is a reduce ring if and only if R is a NZI n-regular ring if and only if R is a left WZI n-regular ring. It is known that clean ring R must be exchange ring, but the converse is still a open question, unless R is an Abelian ring given by Yu(1995a) and Nicholson(1977)) or left quasi-duo ring given by Yu(1995b)). In the chapter, we proof that if R is a NZI ring and an exchange ring, then R is a clean ring.