Certain Rings Related To Prime Radicals

In this thesis,we studied the properties of certain subclasses of clean rings related to prime radicals.We shall combine clean rings,quasipolar rings and Drazin inverse,and introduce three new types of rings: P-clean rings,primely polar rings and primely Drazin invertibility.We investigate the structures of such rings and extend various related properties.The innovation of the thesis chiefly comprises:(1)we generalize the properties of clean rings to P-clean rings,and we generalize the properties of 2-nil-clean rings to 2-P-clean rings,then we get the properties of the 2-P-clean ring;(2)we generalize the properties of quasipolar rings to primely polar rings,and then we studied the ring structure;Through the above studies,we obtained a more detailed description of the relative clean property;(3)we extend the properties of Drazin inverse to primely Drazin inverse,and study the forms of Jacobson's Lemma and Cline's formula on the prime Drazin inverse.This paper includes the following sections:In chapter 1,we introduce the background of clean ring class and the following work of this paper.The main results of this paper are also stated.In chapter 2,we introduce some basic definitions involved in this paper,and then list some important conclusions involved this paper.In chapter 3,we introduce P-clean rings and generalize the property of nilclean rings to P-clean rings;the P-clean properties of matrix ring are also discussed.we shall extend the properties of 2-nil-clean rings to 2-P-clean rings and explore various properties of 2-P-clean rings;We prove that a ring is strongly2-P-clean if and only if for any (6 ? ,(63-(6 ? ()if and only if for any (6 ? ,there exist two orthogonal idempotents 0),1)? such that (6-0)+ 1)? ().Furthermore,related results on strongly 2-P-clean matrix subrings are obtained.In chapter 4,we introduce primely polar rings,and the structure of quasipolar ring was extended to primely polar ring.The properties of primely polar ring were studied and the equivalent characterization of primely polar ring was given.We prove that a commutative strongly -regular ring is primely polar and a primely polar ring is strongly -regular.Moreover,we investigate the characterization of primely polar rings in terms of Drazin inverses.In chapter 5,we introduce the primely Drazin inverse,and then extend the known results of Drazin inverse to such new generalized inverse.We consider the Jacobson's Lemma and Cline's formula for the prime Drazin inverse,and prove the equivalence condition of the primely Drazin inverse.Finally,In chapter 6 some problems related to this paper are put forward and the research prospect is prospected.