Properties and examples

is injective. If this map is bijective then the module is called reflexive. For this reason, torsionless modules are also known as semi-reflexive.

• A free module
  Free module
  In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
  
   is torsionless. More generally, a direct sum
  Direct sum
  In mathematics, one can often define a direct sum of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets , together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question...
  
   of torsionless modules is torsionless.
• A submodule of a torsionless module is torsionless. In particular, any projective module
  Projective module
  In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
  
   over R is torsionless; any left ideal of R is a torsionless left module, and similarly for the right ideals.
• Let R be a domain
  Domain (ring theory)
  In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisors. Some authors require the ring to be nontrivial...
  
  . Then any torsionless module is torsion-free
  Torsion-free
  In mathematics, the term torsion-free may refer to several unrelated notions:* In abstract algebra, a group is torsion-free if the only element of finite order is the identity....
  
  .
• If R is a commutative ring
  Commutative ring
  In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
  
   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.
• Since the rational numbers Q is a divisible abelian group, any Z-linear functional from Q to Z is zero. Thus Q is a torsion-free Z-module which is not 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 (similarly, any right R-module that is a dual of a left R-module is torsionless).

Relation with semihereditary rings

Stephen Chase proved the following characterization of semihereditary rings.

For any ring R, the following conditions are equivalent:

• R is left semihereditary.
• All torsionless right R-modules are flat
  Flat module
  In Homological algebra, and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original...
  
  .
• The ring R is left coherent
  Coherent ring
  In mathematics, a coherent ring is a ring in which every finitely generated left ideal is finitely presented.Many theorems about finitely generated modules over Noetherian rings can be extended to finitely presented modules over coherent rings....
  
  and satsfies 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.

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