Weighted generalized invertibility in two semigroups of a ring with involution

Abstract

Let $R$ be a ring with an involution and $p\in R$ 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 $AA_{M,N}^{\dagger}R^{m \times m}AA_{M,N}^{\dagger}+I_m-AA_{M,N}^{\dagger}$ and the matrix semigroup $A_{M,N}^{\dagger}AR^{n \times n}A_{M,N}^{\dagger}A+I_n-A_{M,N}^{\dagger}A$, where $A\in R^{m \times n}$ be weighted Moore-Penrose invertible with weights $(M,N)$.