Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

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

Author(s): 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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