Torsionless module

In abstract algebra, a module M over a ring R is called torsionless if it can be embedded into some direct product RI. Equivalently, M is torsionless if each non-zero element of M has non-zero image under some R-linear functional f:

This notion was introduced by Hyman Bass.

Properties and examples

A module is torsionless if and only if the canonical map into its double dual,

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

Relation with semihereditary rings

Stephen Chase proved the following characterization of semihereditary rings in connection with torsionless modules:

For any ring R, the following conditions are equivalent:[4]

(The mixture of left/right adjectives in the statement is not a mistake.)

See also

References

  1. P. C. Eklof and A. H. Mekler, Almost free modules, North-Holland Mathematical Library vol. 46, North-Holland, Amsterdam 1990
  2. Proof: If M is reflexive, it is torsionless, thus is a submodule of a finitely generated projective module and hence is projective (semi-hereditary condition). Conversely, over a Dedekind domain, a finitely generated torsion-free module is projective and a projective module is reflexive (the existence of a dual basis).
  3. Bourbaki & Ch. VII, § 4, n. 2. Proposition 8.
  4. Lam 1999, p 146.
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