Unit-Regularity of Regular Nilpotent Elements

TL;DR

This paper investigates conditions under which regular nilpotent elements in rings are unit-regular, providing new decompositions and simplified proofs for properties of strongly π-regular rings and elements.

Contribution

It establishes new decompositions for regular elements under certain conditions and offers simplified proofs for properties of strongly π-regular rings and elements.

Findings

Decomposition of rings involving regular elements and their powers

Proof that strongly π-regular rings have stable range one

Strongly π-regular elements with regular powers are unit-regular

Abstract

Let $a$ be a regular element of a ring $R$. If either $K:=\rm{r}_R(a)$ has the exchange property or every power of $a$ is regular, then we prove that for every positive integer $n$ there exist decompositions $$ R_R = K \oplus X_n \oplus Y_n = E_n \oplus X_n \oplus aY_n,$$ where $Y_n \subseteq a^nR$ and $E_n \cong R/aR$. As applications we get easier proofs of the results that a strongly $\pi$-regular ring has stable range one and also that a strongly $\pi$-regular element whose every power is regular is unit-regular.