Some New Characterizations Of Strongly EP Elements

In an involution ring R with identity element,if the element a is both a PI element and an EP element,then a is called a strongly EP element,or SEP element.In order to further study SEP element,several new characterizations of SEP element are obtained in this paper.In the second chapter,SEP elements are characterized by the constructed invertible elements,and the main conclusions are as follows:(1)a?RSEP if and only if(aa+a-aa*+1-as+)-1=(a#)*a+a+1-aa+;(2)a ? RSEP if and only if 1-aa++a*aa*?R-1 and(1-aa+a*aa*)-1=1-aa++(a+)*aa+;(3)a?RSEP if and only if(a+)*aa+=(a+)*a#(a+)*;if and only if(a+)*aa+=(a#)-a+(a#)*.In the third chapter,some constants are quantified according to known properties or new properties of certain generalized inverse elements in this paper,so as to construct single variable equations or bivariate equations,and then study the corresponding relationship between the constructed equations in a given set ?a or ?a2 in which there are solutions and the element a is a SEP elements.This is a new method,and the following results are obtained:(4)a?RSEP if and only if the equation a#ax=xaa*has at least one solution in ?a;(5)a ? RSEP if and only if the general solution of the equation a#ax=yaa*is given by#12(6)a ? RSEP if and only if the equation a#ax=yaa*and a#ax=yaa#have the same solution;(7)a ? RSEP if and only if the bivariate equation ya*xa+=a+yxa#has at least one solution in ?a2.In the fourth chapter,the(b,c)-inverse of elements is studied to characterize SEP element,the results are listed as follows:(8)a ? RSEP if and only if a+?(a,(a+)*)=(a#)*;(9)a ? RAEP if and only if a*?(a,(a#)*)=a.