On The Г_αβ-Generalized Inverse Of Matrices Over Rings

This paper studies the necessary and sufficient conditions for existence and the explicit expressions of theΓ_αβ-Moore-Penrose inverse over rings with four parts.In the first section, first, we give the definition and the proof of the uniqueness of theΓ_αβ-Moore-Penrose inverse; then we study theΓ_αβ-Moore-Penrose inverse over rings, one of the necessary and sufficient conditions for existence of A_α,β⁺ is given. One of the necessary and sufficient conditions for existence of A_α,β⁺ is the equations XA^*β^*βA = A and Aαα^* A^*Y = A are both soluble. Then B ,C are the solutions of the equations XA^*β^*βA = A and Aαα^*A^*Y = A separately.In the second section, we study theΓ_αβ- {i , ..., j} inverse of a kind of product matrices A = GDH over rings. We give the equivalent conditions for existence and the explicit expressions of theΓ_αβ- {i , ..., j} inverse of a kind of product matrices A = GDH, and some previous results are extended.In the third section, we study theΓ_αβ- Moore-Penrose inverse of a kind of product matrices A = GDH over rings. We give the equivalent conditions for existence and the explicit expressions of theΓ_αβ-Moore-Penrose inverse of a kind of product matrices A = GDH, and some previous results are extended.In the fourth section, we make a further study of theΓ_αβ-generalized inverse of a kind of product matrices A = GDH over rings. We give the necessary and sufficient conditions for existence of theΓ_αβ-generalized inverse of a kind of product matrices A = GDH when D is a general matrix, and some previous results are extended.