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.