The One-Sided Inverse Along an Element in Semigroups and Rings

Abstract

The concept of the inverse along an element was introduced by Mary in 2011. Later, Zhu et al. introduced the one-sided inverse along an element. In this paper, we first give a new existence criterion for the one-sided inverse along a product and characterize the existence of Moore–Penrose inverse by means of one-sided invertibility of certain element in a ring. In addition, we show that $$a\in S^{\dagger }\bigcap S^{\#}$$ if and only if $$(a^{*}a)^{k}$$ is invertible along a if and only if $$(aa^{*})^{k}$$ is invertible along a in a $$*$$ -monoid S, where k is an arbitrary given positive integer. Finally, we prove that the inverse of a along $$aa^{*}$$ coincides with the core inverse of a under the condition $$a\in S^{\{1,4\}}$$ in a $$*$$ -monoid S.