The Generalizations Of Semiregular Rings And Strongly π-regular Rings

Von Neumann regular ring is an important ring. It has attracted many mathematicians since it was introduced. There are many important rings growing out of von Neumann regular rings such as π-reguar rings, strongly π-reguar rings, semiregular rings and so on. It is well known that semiregular rings and strongly π-reguar rings are both semi-π-reguar rings. In this paper, we introduce two classes of rings which are close to regular rings, and give some characterizations of them.In the second chapter, I-semi-π-regular ring is defined. Several characterizations of this ring are given, some results of Nichoson and Zhou are generalized. And relations between I-semi-π-regular and I-semiregular are explored. Let I be some special ideals, we can get some results about these I-semi-π-regular rings. Finally, sufficient and necessary conditions for endomorphism rings of some special modules to be semi-π-regular are given. Some known results are generalized as corollaries.In the third chpter, we combine local rings with semiregular rings and strongly π-regular rings, respectively. Local semiregular rings and local strongly π-regular rings are introduced. Several characterizations of these rings are given. And some results of local rings, semiregular rings, strongly π-regular rings are inhereted to these rings.