Some Applications Of Idempotents In Ring Theory

Exchange rings was first introduced by Prof Warfield in 1972 and a profound result was showed by he, that is, every projective module over an exchange ring is a direct sum of its cyclic submodules. In 1977, Prof Nicholson, an known algebraist of Canada, studied systematically the properties of exchange rings and gave a very classic characterization on exchange rings, that is, a ring R is an exchange ring if and only if idempotents can be lifted modulo every left (right) ideals of R. In order to discussing the properties further, in the same article, he introduced clean rings and showed that clean rings are exchange. But as far as it appears to be an open question whether exchange rings are exactly clean rings in general. This leads to a rather interesting of algebraists, so, reseaching the conditions for Exchange rings R to be clean rings is an important part of algebraic ring-theoretic research. A well-known condition was given by Prof. Nicholson in 1977, that is, R must be an Abelian ring. Later, it was extended to larger rings-left quasi-duo rings by Prof. Yu Huaping in 1995. In 2007. semiabelian rings are introduced by Prof Chen wei-xing. By verifying, one can find that semiabelian rings are one of conditions. Since Clean rings, exchange rings and semiabelian rings are characterized by idempotents, it is an initial motivation of this Doctoral dissertation to clarify the relationship of exchange rings and clean rings.By tracing references, rings having stable range 1 seems to be given by prof Vaserstein in his article:<< Stable rank of rings and dimensionality of topological spaces, Funct. Ana. Appl.5(1971):102-110>>. In 1995, Prof. Yu found that Abelian Exchange rings have stable range 1 and left quasi-duo Exchange rings also have stable range 1. In 2007, Prof Chen wei-xing implied that semiabelian exchange rings have stable range 1. In view of the above-mentioned facts, This Doctoral dissertation natu-rally consider the conditions of exchange rings have stable range 1 by idempotents.In 2006, Prof. Kim Jin Yong proved the following theorem:supposing R is a left hereditary ring with an injective maximal left ideal(called left HI-ring), and if R is a left idempotent reflexive ring, R is a semisimple Artinian Ring. It has denied the question put forward by Prof. Yue Chi Ming in 1989:if a left hereditary ring contains an injective maximal left ideal, is R a semisimple Artinian ring? As an important gen-eralization of Abelian rings, left idempotent reflexive rings gradually pay attention to algebraist. In this Doctoral dissertation, by using the concept of left minimal idempo-tent elements, left MC2 rings are introduced, which are proper generalization of left idempotent reflexive rings and some known results over left idempotent reflexive rings are extended to left MC2 rings.In short, using idempotents, this Doctoral dissertation mainly define the following new rings:Weakly-abel rings, Quasi-normal rings, left EQN rings, left NQD rings; Especially, left minimal idempotents are studied profoundly and using left minimal idempotents, left min-abel rings, left MC2 rings and strong left DS rings are given. In terms of the construction way of idempotents, we study the important properties of rings mentioned above, find out relations not only among them but between them and other rings and using them to extend some existing famous conclusions. We've got results as follows:(1) Weakly-abel exchange rings are clean, which generalizes the result of Prof. Nicholson:Abelian exchange rings are clean and the result of Prof. Yu:left quasi-duo exchange rings are clean.(2) Weakly-abel exchange rings have stable range 1, which generalizes the result owing to Prof. Yu:Abelian or left quasi-duo exchange ring have stable range 1.(3) Left MC2 left HI-rings are semisimple Artinian, which generalizes the result owing to Prof Kim:left idempotent reflexive left HI rings are semisimple Artinian.(4) R is a left quasi-duo ring if and only if R is a left min-abel MELT ring.(5) R is an Abelian ring if and only if R is a quasi-normal left idempotent reflexive ring.(6) R is a strongly regular ring if and only if R is a von Neumann regular quasi-normal ring.(7) R is a strongly left min-abel ring if and only if R is a left MC2 left min-abel ring.(8) R is a strongly left DS ring if and only if R is a left min-abel left universally mminjective ring.