A new class of generalized inverses in semigroups and rings with involution

Abstract

Let S be a $*$-semigroup and let $a,w,v\in S$. 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 $a\in S$ is w-core invertible if there exists some $x\in S$ such that $aw{x}^2=x$, xawa = a and ${(\mathit{awx})}^{*}=\mathit{awx}$. 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 $y\in S$ satisfying $y^{2}va=y$, avay = a and ${(\mathit{yva})}^{*}=\mathit{yva}$. Dual results for the dual v-core inverse also hold.

Communicated by Pace Nielsen

KEYWORDS:

• Core inverses
• dual-core inverses
• Jacobson pairs
• Moore-Penrose inverses
• semigroups with involution
• the inverse along an element
• $\{1,3\}$-inverses
• $\{1,4\}$-inverses

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 15A09
• 16W10
• 16E50

Acknowledgments

The authors are grateful to the referee for his/her careful reading and valuable comments which led to the improvement in this paper. It was notably suggested by the referee to divide the article into three parts, each one discussing a different algebraic structure. Some other results, such as Proposition 2.16 (ii) $\iff$ (iii) and the formulae, Remark 2.27 were given by the referee. Also, the proof of Theorem 3.1 is simplified by the referee.

Additional information

Funding

This research is supported by the National Natural Science Foundation of China (No. 11801124, No. 12171083) and China Postdoctoral Science Foundation (No. 2020M671068).