Strongly One-Sided ( b,c ) -Inverses in Rings

We give some new characterizations of strongly one-sided [Formula: see text]-invertibility of elements in rings, and establish the relations between one-sided [Formula: see text]-inverses and strongly one-sided [Formula: see text]-inverses. When [Formula: see text], Kaplansky's theorem on the number of one-sided inverses is generalized to the case of strongly one-sided [Formula: see text]-inverses. Moreover, we give some new characterizations of Dedekind-finite rings from the viewpoint of [Formula: see text]-finiteness, and thus answer a question in [L. Wang, D. Mosić, The one-sided inverse along two elements in rings, Linear Multilinear Algebra 69 (13) (2021) 2410–2422].