Hybrid (b, c)-inverses and five finiteness properties in rings, semigroups, and categories

Abstract

Given elements a, b, c of any associative ring R with 1, then a is called right hybrid (b, c)-invertible if there exists such that and rann It is shown that such y exists if and only if and in which case y is unique. With an appropriate generalization of the right annhilator rann (.), this result is extended to all semigroups with 1 and to all categories. The semigroup version of right (and left) annihilator ideals is also used to define five finiteness properties for semigroups, and to establish the implications between them. Corresponding results for categories are also obtained.