On *-DMP inverses in a ring with involution

Abstract

An element a in a *-ring R has *-DMP inverse if there exists $m\in \mathbb{N}$ such that $a^m\in R^{\#}\cap R^{†}$ and ${(a^m)}^{\#}={(a^m)}^{†}.$ If m = 1, then a is said to have *-gMP inverse. In this paper, we present several new characterizations of *-DMP inverses in a *-ring. We prove that a has *-DMP inverse if and only if there exists a projection $e\in \mathit{comm}(a)$ such that $a+e\in R^{-1}$ and ae is nilpotent if and only if there exist $m\in \mathbb{N}$ and $x\in R$ such that $x{a}^{m+1}=a^m$ and $ax=xa={(ax)}^{*}.$ Moreover, we prove that $a\in R$ has *-gMP inverse if and only if $a{a}^{*}=ua,a^{*}a=av$ for some $u,v\in R^{-1}.$ The corresponding new characterization of *-DMP inverse is thereby obtained.

Keywords:

• *-DMP inverse
• *-gMP inverse
• projection
• *-ring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 15A09
• 47A11

Acknowledgement

The authors would like to thank the referee for his/her helpful suggestions for the improvement of this paper.

Additional information

Funding

This research was supported by the Natural Science Foundation of Zhejiang Province, China (No. LY21A010018), China Postdoctoral Science Foundation (No. 2020M671281), Research Project of Hubei Provincial Department of Education (No. B2019128).