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arXiv · 1604.05740

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

Bohdan Zabavsky
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Abstract

This article deals mostly with the following question: when is the classical ring of quotients of a commutative ring 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. A commutative any 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 is a (von Neumann) regular local ring if and only if R is a commutative semihereditary local ring.

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Bohdan Zabavsky. 2016-04-19. Type conditions of stable range for identification of qualitative generalized classes of rings. https://arxiv.org/abs/1604.05740

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