
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 Received: 10.07.2017
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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 144152
\mathnet{http://mi.mathnet.ru/adm677}
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