Characterizations and representations of outer inverse for matrices over a ring
Yuanyuan Ke, Jianlong Chen, Predrag S. Stanimirović, Miroslav Ćirić · Linear and Multilinear Algebra · 2019
In this paper, we investigate the existence and representations of the generalized inverse AT,∗(2), AT,∗(1,2), A∗,S(2), A∗,S(1,2), AT,S(2), AT,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 AT,S(2) and AT,S(1,2) of a matrix A by means of {1}-inverse are also given. Furthermore, we present the relationship between the generalized inverse AT,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.