Abstract:
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.
