Narrow Results

A right R-module M_R over any ring R with 1 is called torsion-free if it satisfies the equality 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.

In this paper, by taking the class of all $C3$ (or $D3$) right $R$-modules for general envelopes and covers, we characterize a semisimple artinian ring (or a right perfect ring) via $D3$-covers (or $D3$-envelopes) and a right $V$-ring (or a right noetherian $V$-ring) via $C3$-covers (or $C3$-envelopes). By using isosimple-projective preenvelope, we obtained that if $R$ is a semiperfect right FGF ring (or left coherent ring), then every isosimple right $R$-module has a projective cover. Moreover, we also characterize semihereditary serial rings (respectively, hereditary artinian serial rings) in terms of epic flat (respectively, projective) envelopes.