The Research On Mary Inverses And Related Generalized Inverses

In this dissertation,we investigate the Mary inverse and the weighted generalized inverse in rings.The main results are as follows:The first part mainly studies the Mary inverse of the element and its polarity in rings.We introduce the definition of the polarity along an element,i.e.for any two elements a,b,an element a is polar along d,if there exists some p such that p2=p∈ comm(da),pd=d and da+1-p is a unit.As special cases,the simply polar element,the polar element and the well-supported element belong to the polarity along an element.Then we show that the polar element along an element is equivalent to the existence of the Mary inverse.Finally,we discuss the additivity of the Mary inverse.Second part,we investigate several properties and characterizations of idempotents generated by weighted generalized inverses and weighted pseudo core inverses in*-rings.Several new characterizations about weighted EP elements are given.Finally,we give the existence criteria of weighted pseudo core inverses,and some necessary and sufficient conditions are given to such that the absorption law hold for weighted pseudo core inverses.