Stable range conditions for abelian and duo rings

Abstract

The article deals with the following question: when does the classical ring of quotientsof a duo ring exist and idempotents in the classical ring of quotients $Q_{Cl} (R)$ are thereidempotents in $R$? In the article we introduce the concepts of a ring of (von Neumann) regularrange 1, a ring of semihereditary range 1, a ring of regular range 1. We find relationshipsbetween the introduced classes of rings and known ones for abelian and duo rings.We proved that semihereditary local duo ring is a ring of semihereditary range 1. Also it was proved that a regular local Bezout duo ring is a ring of stable range 2. In particular, the following Theorem 1 is proved: For an abelian ring $R$ the following conditions are equivalent:$1.$\ $R$ is a ring of stable range 1; $2.$\ $R$ is a ring of von Neumann regular range 1. The paper also introduces the concept of the Gelfand element and a ring of the Gelfand range 1 for the case of a duo ring. Weproved that the Hermite duo ring of the Gelfand range 1 is an elementary divisor ring (Theorem 3).