A note on completeness in the theory of strongly clean rings

TL;DR

This paper proves that in a ring complete with respect to an ideal, elements with strongly π-regular images in the quotient are strongly clean in the ring, extending understanding of strong cleanness under ring completion.

Contribution

It establishes a new condition linking strong π-regularity in quotients to strong cleanness in complete rings, expanding the theory of strongly clean rings.

Findings

Elements with strongly π-regular images are strongly clean in complete rings.

Completeness with respect to an ideal influences strong cleanness properties.

The result generalizes previous work on ring extensions and strong cleanness.

Abstract

Many authors have investigated the behavior of strong cleanness under certain ring extensions. In this note, we prove that if $R$ is a ring which is complete with respect to an ideal $I$ and if $x$ is an element of $R$ whose image in $R/I$ is strongly $\pi$-regular, then $x$ is strongly clean in $R$.