Weighted Moore-Penrose Inverse Of Matrices Over Rings

The weighted Moore-Penrose inverse is a special generalized inverse which is an important basic tool in many branches of mathematics and has extensive applications in other fields such as economics, information processing, automatic control, etc. Recently, the study of generalized inverse of matrices associating with more general algebra such as ring, semiring and so on generalize the theory and methods of generalized inverse deeply and characterize the structure of algebra as well.In this paper, we'll give a necessary and sufficient condition for the existence of the weighted Moore-Penrose inverse of a matrix over a ring and generalize it on a semiring. In addition, we'll discuss the relationship between the weighted Moore-Penrose inverse of a matrix and its bording and give the determinantal expression of the weighted Moore-Penrose inverse.