Research On Quasi-Potent Elements And Their Applications In Ring Theory

Ring theory is one of the main research fields of algebra.It has been a hot topic in recent twenty years to describe the structure and properties of rings by decomposing the elements of rings into the sum of some special elements.These special elements mainly include reversible element(unit),idempotent,n-potent element,nilpotent element,etc.In this paper,we firstly introduced the concept of quasi-potent element,which unites the concept of potent ele-ment and the concept of quasi-idempotent.By use of quasi-potent element,we defined quasi-potent ring,semi-quasi-potent ring,quasi-quasi-polar ring and strongly quasi-nil-quasi-clean ring,etc.and a number of examples of rel-ative rings are constructed.A series of characterizations of these new rings are given and many results on Boolean rings,quasi-Boolean rings,potent rings,semi-potent rings,quasi-polar rings and strongly quasi-nil-clean rings are generalized.The main research contents of this paper are the following three parts:In the first part,the concepts of quasi-potent element and quasi-potent ring are defined.An element a in a ring R is called a quasi-potent element if there is a central unit k in R and a positive integer n>1 such that a~n=ka and we say R is called a quasi-potent ring if every element of R is quasi-potent.We study the properties of quasi-potent and quasi-potent rings and got a sur-prising result:a ring is quasi-potent if and only if it is quasi-Boolean.In the second part,the concept of semi-quasi-potent rings is derived by combining quasi-potent elements with Jacobson radical according the semi-potent ring defined by Kos?an,and the related conclusions on semi-potent rings are generalized.For example,we proved that a ring is semi-quasi-potent(semi-quasi-n-potent)if and only if R/J(R)is quasi-potent(quasi-n-potent)and quasi-potent elements(quasi-n-potent elements)lifted modulo J(R).It is shown that a ring R is a semi-quasi-potent ring if and only if ring R is a semi-quasi-n-potent ring for some positive integer n>1 if central units lifted modulo Jacobson radical of R.In the third part,we mainly study quasi-quasi-polar rings and strong-ly quasinil quasi-clean rings.Quasi-quasi-polar ring and strongly quasinil quasi-clean ring are natural generalizations of quasi-polar ring and strongly quasinil clean ring respectively.We proved that quasi-quasi-polar ring and quasi-polar ring are equivalent,every strongly nil quasi-clean ring is strong-ly quasinil quasi-clean ring and every strongly quasinil quasi-clean ring is quasi-polar.