### Abstract

A ring R is said to be strongly π-regular if for every a ∈ R there exist a positive integer n and b ∈ R such that an = an+1b. For example, all algebraic algebras over a field are strongly π-regular. We prove that every strongly π-regular ring has stable range one. The stable range one condition is especially interesting because of Evans' Theorem, which states that a module AI cancels from direct sums whenever EndR(M) has stable range one. As a consequence of our main result and Evans' Theorem, modules satisfying Fitting's Lemma cancel from direct sums. ©1996 American Mathematical Society.

Original language | English |
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Pages (from-to) | 3293-3298 |

Journal | Proceedings of the American Mathematical Society |

Volume | 124 |

Issue number | 11 |

Publication status | Published - 1 Dec 1996 |

### Keywords

- Exchange ring
- Fitting's Lemma
- Stable range one
- Strongly π-regular ring

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## Cite this

Ara, P. (1996). Strongly π-regular rings have stable range one.

*Proceedings of the American Mathematical Society*,*124*(11), 3293-3298.