Separative exchange rings in which 2 is invertible

TL;DR

This paper studies properties of separative exchange rings with invertible 2, focusing on conditions for elements to be unit-regular or special clean, extending known results in regular rings.

Contribution

It establishes new criteria for unit-regularity and special cleanness of elements in separative exchange rings with invertible 2, generalizing previous results in regular rings.

Findings

Characterization of unit-regular elements under specific ideal conditions

Conditions for elements to be special clean involving projectivity

Extension of results from regular rings to separative exchange rings

Abstract

An exchange ring $R$ is separative provided that for all finitely generated projective right $R$-modules $A$ and $B$, $A\oplus A\cong A\oplus B\cong B\oplus B\Longrightarrow A\cong B$. Let $R$ be a separative exchange ring in which $2$ is invertible, and let $a-a^3\in R$ be regular. We prove, in this note, that $a\in R$ is unit-regular if $R(1-a^2)R=Rr(a)={\ell}(a)$. An element $a$ in a ring $R$ is special clean if there exists an idempotent $e\in R$ such that $a-e\in R$ is a unit and $aR\bigcap eR=0$. Furthermore, we prove that $a\in R$ is special clean if $aR/ar(a^2), R/\big(aR+r(a)\big)$ are projective, and $R(a-a^3)R=Rar(a^2)=\ell (a^2)aR$. These also extend the corresponding results in separative regular rings.