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Unit-Regularity of Regular Nilpotent Elements
- Publication Year :
- 2015
-
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.<br />Comment: In the revision some typos are corrected, minor modifications are made and also references to two related papers are added. To appear in Algebras and Representation Theory
- Subjects :
- Mathematics - Rings and Algebras
16E50
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1509.07944
- Document Type :
- Working Paper