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

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.