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

Bohdan Zabavsky

Algebra and discrete mathematics, Tome 26 (2018) no. 1, pp. 144-152

Résumé

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.

Détail

Keywords: Bezout ring, Hermite ring, elementary divisor ring, semihereditary ring, regular ring, neat ring, clean ring, stable range 1.