Generalized Inverse Of Partitioned Morphisms

In this article, the Weighted Moore-Penrose inverse of matrices over rings, the WeightedΓ-inverse of matrices, the weighted Moore-Penrose inverse of morphisms, the weighted (i , , j ) inverse of morphisms, the weighted Moore-Penrose inverse of morphisms with kernels and its Core-Nilpotent Decomposition, the generalized inverse of partitioned morphisms are studied. The main results are listed as follows:(1)The Weighted Moore-Penrose inverse of A = GDH over rings with respect to M and N is studied, the necessary and sufficient conditions of its existence are given. Some previous results on the weighted generalized inverse and generalized inverse are enhanced.(2)The WeightedΓ-inverse of matrices is studied. the minimum-norm M solution, the least-squares M solution, the minimum-norm M and least-squares M solution of linear equations APx = b are given. Some results on theΓ-inverse of matrices are extended. (3) The necessary and sufficient conditions for existences and expressions of the weighted Moore-Penrose inverse of morphisms with universal-factorization in the preadditive category are given. Some previous results on Moore-Penrose inverse of morphisms with universal-factorization are extended.(4) The weighted inverse of morphisms with universal-factorization in the preadditive category are researched, and some new necessary and sufficient conditions for existences and expressions of it are given. Some previous results on the generalized (i , , j )inverse of morphisms with universal-factorization are extended.(5)w-weighted Drazin inverse of morphisms with kernels and its Core-Nilpotent Decomposition in an additive category is discussed, the Drazin inverse of morphisms with kernels and its Core-Nilpotent Decomposition are enhanced, the representations of the w-weighted Drazin inverse of morphisms with kernels are obtained.(6) the extension of Cline partitioned morphism is introduced and formulas for its Moore-Penrose inverse and Drazin inverse and Group inverse are derived.