Strongly $\pi$-regular rings have stable range one
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Abstract:
A ring $R$ is said to be strongly $\pi$-regular if for every $a\in R$ there exist a positive integer $n$ and $b\in R$ such that $a^{n}=a^{n+1}b$. For example, all algebraic algebras over a field are strongly $\pi$-regular. We prove that every strongly $\pi$-regular ring has stable range one. The stable range one condition is especially interesting because of Evans’ Theorem, which states that a module $M$ cancels from direct sums whenever $\text {End}_{R} (M)$ has stable range one. As a consequence of our main result and Evans’ Theorem, modules satisfying Fitting’s Lemma cancel from direct sums.References
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Additional Information
- Pere Ara
- Affiliation: Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
- MR Author ID: 206418
- Email: para@mat.uab.es
- Received by editor(s): April 28, 1995
- Additional Notes: The author was partially supported by DGYCIT grant PB92-0586 and the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.
- Communicated by: Ken Goodearl
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 3293-3298
- MSC (1991): Primary 16E50, 16U50, 16E20
- DOI: https://doi.org/10.1090/S0002-9939-96-03473-9
- MathSciNet review: 1343679