A new class of generalized inverses in semigroups and rings with involution
Huihui Zhu, Liyun Wu, Jianlong Chen · Communications in Algebra · 2022
Let S be a *-semigroup and let a,w,v∈S. The initial goal of this work is to introduce two new classes of generalized inverses, called the w-core inverse and the dual v-core inverse in S. An element a∈S is w-core invertible if there exists some x∈S such that awx2=x, xawa = a and (awx)*=awx. Such an x is called a w-core inverse of a. It is shown that the core inverse and the pseudo core inverse can be characterized in terms of the w-core inverse. Several characterizations of the w-core inverse of a are derived, and the expression is given by the inverse of w along a and {1, 3}-inverses of a in S. Also, the connections between the w-core inverse and other generalized inverses are given. In particular, when S is a *-ring, the criterion for the w-core inverse is given by units. The dual v-core inverse of a is defined by the existence of y∈S satisfying y2va=y, avay = a and (yva)*=yva. Dual results for the dual v-core inverse also hold.Communicated by Pace Nielsen