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

• Date: 2018
• Main Author: Zabavsky, Bohdan Volodymyrovych
• Format: Article
• Language: English
• Published: Lugansk National Taras Shevchenko University 2018
• Subjects: Bezout ring; Hermite ring; elementary divisor ring; semihereditary ring; regular ring; neat ring; clean ring; stable range 1; 13F99; 06F20
• Online Access: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/503
• Journal Title: Algebra and Discrete Mathematics

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.