Characterizations of weighted (b, c) inverse

The notion of weighted $(b,c)$-inverse of an element in rings were introduced, very recently [Comm. Algebra, 48 (4) (2020): 1423-1438]. In this paper, we further elaborate on this theory by establishing a few characterizations of this inverse and their relationships with other $(v, w)$-weighted $(b,c)$-inverses. We introduce some necessary and sufficient conditions for the existence of the hybrid $(v, w)$-weighted $(b,c)$-inverse and annihilator $(v, w)$-weighted $(b,c)$-inverse of elements in rings. In addition to this, we explore a few sufficient conditions for the reverse-order law of the annihilator $(v, w)$-weighted $(b,c)$-inverses.

[1]  R. Duffin,et al.  On the algebra of networks , 1953 .

[2]  M. Drazin.  A class of outer generalized inverses , 2012 .

[3]  V. Katsikis,et al.  Representations and properties of the W-Weighted Drazin inverse , 2017 .

[4]  X. Mary.  On generalized inverses and Green's relations , 2009, 0903.1785.

[5]  Jerry J. Koliha,et al.  Moore–Penrose inverse in rings with involution , 2007 .

[6]  Dragan S. Raki'c,et al.  Group, Moore–Penrose, core and dual core inverse in rings with involution☆ , 2014, 1403.8108.

[7]  Huihui Zhu.  Further results on several types of generalized inverses , 2018 .

[8]  T. Greville.  Note on the Generalized Inverse of a Matrix Product , 1966 .

[9]  Chunyuan Deng,et al.  On a weighted core inverse in a ring with involution , 2018 .

[10]  Mengmeng Zhou,et al.  Weak group inverses in proper ∗-rings , 2020 .

[11]  Invertibility of the Difference of Idempotents , 2003 .

[12]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[13]  V. Katsikis,et al.  Core and core-EP inverses of tensors , 2019, Computational and Applied Mathematics.

[14]  M. Drazin.  Weighted (b,c)-inverses in categories and semigroups , 2020 .

[15]  R. Hartwig,et al.  A note on the group structure of unit regular ring elements , 1977 .

[16]  Sourav Das,et al.  Further results on weighted core inverse in a ring , 2020, Linear and Multilinear Algebra.

[17]  M. Drazin.  Pseudo-Inverses in Associative Rings and Semigroups , 1958 .

[18]  M. Drazin.  Left and right generalized inverses , 2016 .

[19]  M. Drazin.  Commuting properties of generalized inverses , 2013 .