EP,Normal Elements And Generalized Partial Isometries In Rings

EP elements,normal elements and partial isometries are widely used in many fields,and a number of scholars extensively studied generalized inverses of complex matrices,bounded linear operators on Banach spaces,Banach algebras,C*-algebras,rings and semigroups.However,there are still some problems to study.This paper mainly focuses on the characterizations of EP elements,normal elements and generalized partial isometries in rings.It mainly consists of three parts.The first part mainly focuses on the characterizations of EP elements under some con-ditions,First of all,we discuss the equivalent conditions of EP elements in rings involving their core inverses.For example,let a?R(?),then a is EP if and only if(a(?))2a#=a(?)a#a(?).This generalizes D.Mosie's corresponding results.Also,we present some characterizations of EP elements in rings involving powers of their MP-inverses and group inverses.For instance.let a?R(?)?R#,n?N,then a is EP if and only if a#(a(?))n+1=a(?)(a#)na(?).This generalizes D.Mosie's corresponding results.Finally,we present the equivalent conditions of elements within epic-monic factorization to be EP.The second part mainly considers the characterizations of normal elements,generalized normal elements and hermitian elements under some conditions.On the one hand,we present some equivalent conditions of normal elements in rings involving their core inverses.For example,let a?R(?),then a is normal if and only if a*a(?)=a(?)a*.This generalizes D.Mosie's corresponding results.On the other hand,we give the equivalent conditions of normal,generalized normal and hermitian elements in rings involving powers of their MP-inverses and group inverses.For instance,let a?R(?)?R#,n?N,then a is normal if and only if a*a(?)(a#)n=a#a*(a(?))n.This generalizes D.Mosie's corresponding results.The third part mainly establishes the characterizations of generalized partial isometries and core involutory elelnents under some conditions.Among all the results,we consider the equivalent conditions of generalized partial isometries and EP elements in rings involving their MP-inverses and group inverses.For example,let a ? R(?),n ? N,then a is generalized partial isometry and EP if and only if a?R#,a(a*)n=(a(?))na.This generalizes D.Mosic's corresponding results.Then,we exhibit the equivalent conditions of core involutory elements in rings involving their core inverses.For instance,let a?R(?),then a is core involutory if and only if a*a=a(?)a.