Generalized inverses of morphisms with kernels

The Moore–Penrose inverse of von Neumann regular matrices over a ring

“…Standard book references are [4], [5], [14], [23]. Many authors considered Moore-Penrose invertibility over more general rings (see [2], [8], [10], [11], [12], [15], [17], [19], [24], [25]) and even for morphisms in (additive) categories with involutions (see [20], [21], [22], [26]). For a description of the evolution of generalized invertibility and a complete list of references on the subject up to 1986, the reader is referred to [3].…”

“…AA * P * P AÜ −1 A * AA †V −1 AÜ −1 A * P * using(25) = Q * A † AA * AA †V −1 AÜ −1 A * P * using(26) = Q * A * V −1 AÜ −1 A * P * = (AQ) * V −1 AÜ −1 (P A) * . 2…”

The outer inverse f(2)T,S of a homomorphism of right R-modules

“…More generally, an R−homomorphism of modules is regarded as a morphism in the category of modules, which is an additive category. The Moore-Penrose inverses and other generalized inverses of a morphism in an additive category are studied by many authors (see [4,6,[9][10][11]).…”

Core and dual core inverses of a sum of morphisms

“…(See, [1] and [5]- [9].) In [5], Robinson and Puystjens give the characterizations about the Moore-Penrose inverse and the group inverse of a morphism with kernels. In [6], Miao and Robinson investigate the group and Moore-Penrose inverses of regular morphisms with kernel and cokernel.…”

“…Group inverses and Moore-Penrose inverses of morphisms were investigated some years ago. (See, [1] and [5]- [9].) In [5], Robinson and Puystjens give the characterizations about the Moore-Penrose inverse and the group inverse of a morphism with kernels.…”