2016/04/19 by Bohdan Zabavsky, Zabavsky, Bohdan
Decision Sciences · Mathematics · Computer Science · #Fuzzy and Soft Set Theory #Advanced Topics in Algebra #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.1604.05740
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.