Torsionless module

Properties and examples

• A unital free module is torsionless. More generally, a direct sum of torsionless modules is torsionless.
• A free module is reflexive if it is finitely generated, but for some rings there are also infinitely generated free modules that are reflexive. For instance, the direct sum of countably many copies of the integers is a reflexive module over the integers, see for instance.
• A submodule of a torsionless module is torsionless. In particular, any projective module over R is torsionless; any left ideal of R is a torsionless left module, and similarly for the right ideals.
• Any torsionless module over a domain is a torsion-free module, but the converse is not true, as Q is a torsion-free Z-module which is not torsionless.
• If R is a commutative ring which is an integral domain and M is a finitely generated torsion-free module then M can be embedded into Rⁿ and hence M is torsionless.
• Suppose that N is a right R-module, then its dual N^∗ has a structure of a left R-module. It turns out that any left R-module arising in this way is torsionless.
• Over a Dedekind domain, a finitely generated module is reflexive if and only if it is torsion-free.
• Let R be a Noetherian ring and M a reflexive finitely generated module over R. Then is a reflexive module over S whenever S is flat over R.

  Relation with semihereditary rings

• R is left semihereditary.
• All torsionless right R-modules are flat.
• The ring R is left coherent and satisfies any of the four conditions that are known to be equivalent:
• * All right ideals of R are flat.
• * All left ideals of R are flat.
• * Submodules of all right flat R-modules are flat.
• * Submodules of all left flat R-modules are flat.