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

TL;DR

This paper investigates conditions under which classical rings of quotients of commutative rings have stable range 1, introducing new classes of rings and establishing relationships among them, with implications for ring regularity and cleanness.

Contribution

It introduces new concepts like rings of regular range 1 and explores their relationships with known ring classes, advancing understanding of stable range conditions.

Findings

A commutative indecomposable almost clean ring is a regular local ring.

A ring of idempotent regular range 1 is almost clean.

The classical ring of quotients of a commutative Bezout ring is a regular local ring iff the ring is semihereditary and local.

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.