Characterizations of core and dual core inverses in rings with involution

TL;DR

This paper characterizes the core and dual core inverses in rings with involution using Hermitian elements, projections, and units, providing new insights into their structure and relations to other inverses.

Contribution

It introduces novel characterizations and representations of core and dual core inverses in rings with involution, including conditions involving Hermitian elements and invertibility criteria.

Findings

Characterization of core invertibility via Hermitian elements and invertibility conditions

New criteria for EP elements involving projections and invertibility

Expressions for core and dual core inverses in terms of units and regular elements

Abstract

Let R be a unital ring with involution, we give the characterizations and representations of the core and dual core inverses of an element in R by Hermitian elements (or projections) and units. For example, let a in R and n is an integer greater than or equal to 1, then a is core invertible if and only if there exists a Hermitian element (or a projection) p such that pa=0, a^n+p is invertible. As a consequence, a is an EP element if and only if there exists a Hermitian element (or a projection) p such that pa=ap=0, a^n+p is invertible. We also get a new characterization for both core invertible and dual core invertible of a regular element by units, and their expressions are shown. In particular, we prove that for n is an integer greater than or equal to 2, a is both Moore-Penrose invertible and group invertible if and only if (a*)^n is invertible along a.