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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Strongly $\pi $-regular rings
have stable range one


Author: Pere Ara
Journal: Proc. Amer. Math. Soc. 124 (1996), 3293-3298
MSC (1991): Primary 16E50, 16U50, 16E20
MathSciNet review: 1343679
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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.


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Additional Information

Pere Ara
Affiliation: Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Email: para@mat.uab.es

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03473-9
PII: S 0002-9939(96)03473-9
Keywords: Strongly $\pi $-regular ring, stable range one, exchange ring, Fitting's Lemma
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
Article copyright: © Copyright 1996 American Mathematical Society