The generalized inverse over commutative rings

In this article, we establish the definition of the generalized inverse , which is {2} inverse of matrix A with prescribed image T and kernel S over a commutative ring R, and give an explicit expression for over integral domains, which generalizes the explicit expression for over the field of complex numbers [Wei Yimin, 1998, A characterization and representation of the generalized inverse and its applications, Linear Algebra Applications, 280, 87–96, Theorem 2.1]. In addition, we show that over integral domains, the Drazin inverse, the group inverse and the Moore–Penrose inverse are all .