Rank equalities related to a class of outer generalized inverse
Abstract
In 2012, Drazin introduced a class of outer generalized inverse in a ring R,
the (b; c)-inverse of a for a; b; c in R and denoted by a^(b;c). In this paper, rank equalities
of A^k A^(B;C) - A^(B;C)A^k and (A*)^k A^(B;C)-A^(B;C)(A*)^k are obtained. As application-
s, we investigate equivalent conditions for the equalities (A*^)kA^(B;C) = A^(B;C)(A*)^k and
A^k A^(B;C) = A^(B;C)A^k. As corollaries we obtain rank equalities related to the Moore-Penrose
inverse, the core inverse, and the Drazin inverse. The paper nishes with some rank equalities
involving different expressions containing A^(B;C).
the (b; c)-inverse of a for a; b; c in R and denoted by a^(b;c). In this paper, rank equalities
of A^k A^(B;C) - A^(B;C)A^k and (A*)^k A^(B;C)-A^(B;C)(A*)^k are obtained. As application-
s, we investigate equivalent conditions for the equalities (A*^)kA^(B;C) = A^(B;C)(A*)^k and
A^k A^(B;C) = A^(B;C)A^k. As corollaries we obtain rank equalities related to the Moore-Penrose
inverse, the core inverse, and the Drazin inverse. The paper nishes with some rank equalities
involving different expressions containing A^(B;C).
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