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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2018, Volume 26, Issue 1, Pages 144–152 (Mi adm677)

RESEARCH ARTICLE

Type conditions of stable range for identification of qualitative generalized classes of rings

Bohdan Zabavsky

Department of Mechanics and Mathematics, Ivan Franko National University of L'viv, Lviv, Ukraine

Abstract: This article deals mostly with the following question: when the classical ring of quotients of a commutative ring is a ring of stable range 1? We introduce the concepts of a ring of (von Neumann) regular range 1, a ring of semihereditary range 1, a ring of regular range 1, a semihereditary local ring, a regular local ring. We find relationships between the introduced classes of rings and known ones, in particular, it is established that a commutative indecomposable almost clean ring is a regular local ring. Any commutative ring of idempotent regular range 1 is an almost clean ring. It is shown that any commutative indecomposable almost clean Bezout ring is an Hermite ring, any commutative semihereditary ring is a ring of idempotent regular range 1. The classical ring of quotients of a commutative Bezout ring $Q_{Cl}(R)$ is a (von Neumann) regular local ring if and only if $R$ is a commutative semihereditary local ring.

Keywords: Bezout ring, Hermite ring, elementary divisor ring, semihereditary ring, regular ring, neat ring, clean ring, stable range 1.

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MSC: 13F99, 06F20
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Citation: Bohdan Zabavsky, “Type conditions of stable range for identification of qualitative generalized classes of rings”, Algebra Discrete Math., 26:1 (2018), 144–152

Citation in format AMSBIB
\Bibitem{Zab18} \by Bohdan~Zabavsky \paper Type conditions of stable range for identification of qualitative generalized classes of rings \jour Algebra Discrete Math. \yr 2018 \vol 26 \issue 1 \pages 144--152 \mathnet{http://mi.mathnet.ru/adm677}