Further results on the inverse along an element in semigroups and rings
Huihui Zhu, Jianlong Chen, Pedro Henrique de Bitencourt Patricio · Linear and Multilinear Algebra · 2015
In this paper, we introduce a new notion in a semigroup as an extension of Mary’s inverse. Let . An element is called left (resp. right) invertible along if there exists such that (resp. ) and (resp. ). An existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right) -regularity and left (right) -regularity are given in a semigroup. Further, another existence criterion of this type inverse is given by means of a left (right) invertibility of certain elements in a ring. Finally, we study the (left, right) inverse along a product in a ring, and, as an application, Mary’s inverse along a matrix is expressed.