No abstract

No abstract

In this paper we introduce left and right annihilator (b,c)-inverses and we investigate some of theirs properties. Furthermore, here we study some properties of left and right (b,c)-inverses.

In this paper we introduce a new generalized inverse in a ring -one-sided (b, c)-inverse, derived as an extension of (b, c)-inverse. This inverse also generalizes onesided inverse along an element, which was recently introduced by H. H. Zhu et al. [H. H. Zhu, J. L. Chen, P. Patrício, Further results on the inverse along an element in semigroups and rings, Linear Multilinear Algebra, 64 (3) (2016) 393-403]. Also, here we present one-sided annihilator (b, c)-inverse, which is an extension of the annihilator (b, c)-inverse. Necessary and sufficient conditions for the existence of these new generalized inverses are obtained. Furthermore, we investigate conditions for the existence of one-sided (b, c)inverse of a product of three elements and we consider some properties of one-sided (b, c)inverses.

Representations of 1,2,3-inverses, 1,2,4-inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.