Several Studies Of Hermitian Elements In Rings With Involution

In an involution ring R with involution,if the element a satisfies a＝a*,then a is called a Hermitian element.This thesis mainly uses a constructive method to give several new characterizations of Hermitian element.The first chapter mainly introduces the definitions of the terms used in this thesis,as well as some theorems and lemmas.The second chapter is first portrayed by the definition of Hermitian element to a series of equivalence of its characterization,and then describes Hermitian elements by constructing the invertible element to get the following conclusions:(1)a ∈RHer if and only if a2=a*aa*u,where u∈R-1 and ua+a=(a#)*;(2)a∈RHer if and only if aa+=a*v,a(aa#)*=va2,where v∈R-1.The third chapter defines a for those elements being both collection of the aggressive element that is both group invertible and Moore-Penrose invertible,then construct certain equations such that an element of a ring with involution is Hermitian if and only if these equations have solutions in χa.Next,we characterize Hermitian elements by means of the consistency of some equations.Finally,the equation constructed by generalization and explore its general solution,and get the following conclusions:(3)a∈RHer if and only if the general solution of the equation a*x(aa#)*=a2 is given byx=a+a2+u-aa+ua+a;(4)a∈RHer if and only if the equation a*x(aa#)*=xa has at least one solution in xa;(5)a∈RHer if and only if the bivariate equation a*xa+ya+=xya#has at least one solution in χa2;(6)a∈RHer if and only if the general solution of the bivariate equation a*xa+=yaa#is given by(?),where p,u,v∈R.In fourth chapter,we are studied the required conditions of the Moore-Penrose invertible element and the group invertible element as Hermitian element to obtain the following conclusions:(7)a∈RHer if and only if exists x∈χa,then(a’xa+)#=(aa#)*a(aa#)*x+a#;(8)a∈RHer if and only if exists x∈τa,then(aa#)*a(aa#)*x+(a#)*=aa+x#;(9)a∈RHer if and only if exists x∈χa,then ax#(a+)*=a+ax+.