Torsionless module

In abstract algebra, a module "M" over a ring "R" is called torsionless if it can be embedded into some direct product "R""I". 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: