Strongly π-regular rings have stable range one

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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 languageEnglish
Pages (from-to)3293-3298
JournalProceedings of the American Mathematical Society
Volume124
Issue number11
Publication statusPublished - 1 Dec 1996

Keywords

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

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    Ara, P. (1996). Strongly π-regular rings have stable range one. Proceedings of the American Mathematical Society, 124(11), 3293-3298.