The generalized inverse of two idempotent matrices and their product under the similar transformation over a local ring

Let R is a local ring,the theory of the generalized inverse of one matrix is extended under the sameness similar transformation over R.By using the matrix methods and the theory of the normal form of matrix under the sameness similar transformation,the generalized inverse of two idempotent matrices and their product are discussed under the sameness similar transformation over R,and the following conclusions are given that(1) {k}-inverse of two idempotent matrices and their product under the sameness similar transformation,where k=1,3;(2) {i,j}-inverse of two idempotent matrices and their product under the sameness similar transformation,where i≠j.