Suitable elements, -clean elements and Sylvester equations in rings with involution

Abstract

Let R be a $*$-ring. In this article, we introduce good $*$-clean elements and left $*$-suitable elements in R, respectively. An element $a\in R$ is good $*$-clean if it can be written as the sum of a projection and a unitary element. An element $a\in R$ is left $*$-suitable if there exists a projection $p\in Ra$ such that $1-p\in R(1-a).$ Several properties of them are given and their characterizations are derived by the solvability of the Sylvester equation $XA-BX=C$ in a ring. Finally, we use generalized inverses to give the existence criterion of left $*$-suitable elements.

Keywords:

• $*$-Clean elements
• regularities
• suitable elements
• Sylvester equations

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16E50
• 16U99

Acknowledgments

The authors are highly grateful to the referee for his/her valuable comments on this article.

Additional information

Funding

This research is supported by the National Natural Science Foundation of China (No. 11801124 and 11971294) and China Postdoctoral Science Foundation (No. 2020M671068).