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arXiv · 1509.07944

Unit-Regularity of Regular Nilpotent Elements

Dinesh Khurana
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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.

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Dinesh Khurana. 2015-09-26. Unit-Regularity of Regular Nilpotent Elements. https://arxiv.org/abs/1509.07944

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