MMN-3971

# Pseudo core invertibility and DMP invertibility in two semigroups of a ring with involution

*Wende Li*;

*Jianlong Chen*;

*Yukun Zhou*;

*Yuanyuan Ke*;

## Abstract

In 2004, Patr\'{i}cio and Puystjens characterized the relation between Drazin invertible elements (resp., Moore-Penrose invertible elements) of two semigroups $pRp$ and $pRp+1-p$ of a ring $R$ for some idempotent (resp., projection) $p\in R$. In this paper, we consider the relevant result for pseudo core invertible elements of such two semigroups of a ring for some projection, which is then applied to characterize the relation between pseudo core invertible elements of the matrix semigroup $AA^{\dagger}R^{m\times m}AA^{\dagger}+I_m-AA^{\dagger}$ and the matrix semigroup $A^{\dagger}AR^{n\times n}A^{\dagger}A+I_n-A^{\dagger}A$, where $A\in R^{m \times n}$ with $A^{\dagger}$ existing and $B\in R^{m \times m}$. Also, similar equivalence involving DMP invertible elements is investigated.

Vol. 24 (2023), No. 3, pp. 1427-1438

DOI: 10.18514/MMN.2023.3971