Commutatively closed properties of (b,c)-inverses

This paper extends the study of commutatively closed properties to the case of [Formula: see text]-inverses, building on the foundational work of commutatively closed sets established in [D. Alghazzawi and A. G. Leroy, Commutatively closed sets in rings, Comm. Algebra 47(4) (2019) 1629–1641]. We develop various criteria characterizing the commutatively closed behavior of [Formula: see text]-inverses. In addition, the study is further extended to some specific instances including [Formula: see text]-inverses, group inverses and core inverses, which reveal their distinctive commutatively closed properties. As an application, we characterize the commutatively closed properties of [Formula: see text]-inverses of diagonal matrices.