The one-sided inverse along an element in semigroups and rings
In [Linear Algebra Appl. 434, No. 8, 1836--1844 (2011; Zbl 1219.15007)], \textit{X. Mary} introduced the concept of the inverse along an element. In this paper, the authors give a new existence criterion for the one-sided inverse along a product and characterize the existence of the Moore-Penrose inverse by means of one-sided invertibility of certain elements in a ring. For a \(\ast\)-monoid \(S\) (\(\ast: S\to S\) an involution), they show that \(a\in S^{\dagger}\cap S^\#\) if and only if \((a^\ast a)^k\) is invertible along \(a\) if and only if \((aa^\ast)^k\) is invertible along \(a\) in \(S\). Here, \(k\) is an arbitray positive integer. They also prove that the inverse of \(a\) along \(aa^\ast\) coinsides with the core inverse of \(a\) under the condition \(a\in S^{\{1,4\}}\) in \(S\).
scientific article; zbMATH DE number 6804304
scientific article