Torsion-free module
From Encyclopedia of Mathematics
A module over a ring
without divisors of zero, such that the equality
, where
,
, implies
or
. Examples of such (left) modules are the ring
itself and all its non-zero left ideals. A submodule of a torsion-free module and also the direct sum and direct product of torsion-free modules are torsion-free modules. If
is commutative, then for any module
there is a torsion submodule
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In this case the quotient module is torsion-free.
Comments
More generally, for any associative ring a left
-module
is called torsion-free if for
,
for a regular element
implies
. Cf. Torsion submodule for more details and some references.
How to Cite This Entry:
Torsion-free module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion-free_module&oldid=17214
Torsion-free module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion-free_module&oldid=17214
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article