On (b, c)-inverses and (c, b)-inverses. Issue 10 (4th August 2021)

• Record Type: Journal Article
• Authors: Wu, Cang; Chen, Jianlong
• Is Part Of: Communications in algebra. Volume 49:Issue 10(2021)
• Page Start: 4313
• Page End: 4323
• Publication Date: 2021-08-04
• Subjects: (b, c)-Inverse -- one-sided (b, c)-inverse -- inverse along an element -- group inverse -- weighted Moore-Penrose inverse -- core-EP inverse; 15A09; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2021.1918701
• Languages: English
• ISSNs: 0092-7872

Abstract

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.