TORSION-FREENESS FOR RINGS WITH ZERO-DIVISORS

Abstract

A right R-module MR over any ring R with 1 is called torsion-free if it satisfies the equality [Formula: see text] for every r∈R. An equivalent definition was used by Hattori [11]. We establish various properties of this concept, and investigate rings (called torsion-free rings) all of whose right ideals are torsion-free. In a torsion-free ring, the right annihilators of elements are always idempotent flat right ideals. The right p.p. rings are characterized as torsion-free rings in which the right annihilators of elements are finitely generated. An example shows that torsion-freeness ness is not a Morita invariant. Several ring and module properties are proved, showing that, in several respects, torsion-freeness ness behaves like flatness. We exhibit examples to point out that the concept of torsion-freeness ness discussed here is different from other notions.