TORSION-FREENESS FOR RINGS WITH ZERO-DIVISORS

A right R-module M-R over any ring R with 1 is called torsion-free if it satisfies the equality Tor(1)(R)(M, R/Rr) = 0 for every r is an element of 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 is not a Morita invariant. Several ring and module properties are proved, showing that, in several respects, torsion-freeness behaves like flatness. We exhibit examples to point out that the concept of torsion-freeness discussed here is different from other notions.