On (b, c)-inverses and (c, b)-inverses. Issue 10 (4th August 2021) - Wu, Cang; Chen, Jianlong
Abstract: We extend characterizations of inverses along an element to ( b, c )-inverses. It is proved that if an element a in a semigroup S is both ( b, c )-invertible and ( c, b )-invertible for some b, c ∈ S, then abac, acab, caba, baca are group invertible, and bac ( abac ) ♯ = b a ( caba ) ♯ c = b ( acab ) ♯ a c = ( baca ) ♯ b a c is the ( b, c )-inverse of a, cab ( acab ) ♯ = c a ( baca ) ♯ b = c ( abac ) ♯ a b = ( caba ) ♯ c a b is the ( c, b )-inverse of a . Moreover, we establish several criteria for the existence of ( b, c )-inverses and ( c, b )-inverses by means of units in a ring. As applications, some new representations of weighted Moore-Penrose inverses and core-EP inverses of complex matrices are given.