LAUSR

Let R be a ring with an involution and p ? R be a weighted projection. We characterize the relation between the weighted Moore-Penrose invertibility (resp., weighted pseudo core invertibility) of the corresponding elements of the two semigroups pRp and pRp + 1-p. As an application, we obtain the relation between the weighted Moore-Penrose invertibility (resp., weighted pseudo core invertibility) of the corresponding elements of the matrix semigroup AA? M,N Rm?m AA?M,N + Im-AA?M,N and the matrix semigroup A?M,N ARn?n A?M,N A + In-A?M,N A, where A ? Rm?n be weighted Moore-Penrose invertible with weights (M, N).