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

• Published in: Algebra and Discrete Mathematics
• Date: 2018
• Main Author: Zabavsky, B.V.
• Format: Article
• Language: English
• Published: Інститут прикладної математики і механіки НАН України 2018
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/188381
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Type conditions of stable range for identification of qualitative generalized classes of rings / B.V. Zabavsky // Algebra and Discrete Mathematics. — 2018. — Vol. 26, № 1. — С. 144–152 . — Бібліогр.: 6 назв. — англ.

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 QCl(R) is a (von Neumann) regular local ring if and only if R is a commutative semihereditary local ring.