Characterizations and representations of outer inverse for matrices over a ring. Issue 1 (2nd January 2021) - Ke, Yuanyuan; Chen, Jianlong; Stanimirović, Predrag; Ćirić, Miroslav
Abstract : In this paper, we investigate the existence and representations of the generalized inverse A T, ∗ ( 2 ), A T, ∗ ( 1, 2 ), A ∗, S ( 2 ), A ∗, S ( 1, 2 ), A T, S ( 2 ), A T, S ( 1, 2 ) of a matrix A over an associative ring, and give some characterizations and representations of these generalized inverses by using regularity. Some explicit expressions for A T, S ( 2 ) and A T, S ( 1, 2 ) of a matrix A by means of 1 -inverse are also given. Furthermore, we present the relationship between the generalized inverse A T, S ( 2 ) and the ( B, C ) -inverse of a matrix A over rings. And then using the regularity give the sufficient and necessary condition for the existence and the expression of ( B, C ) -inverse of matrices over a ring. Some explicit expressions for ( b, c ) -inverse of an element in semigroups by means of 1 -inverse are also studied. As applications, some new results for the Mary inverse, the core inverse and dual core inverse are obtained.