Torsion-free module

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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

In this case the quotient module is torsion-free.


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. L.V. Kuz'min (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098