# 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

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. L.V. Kuz'min (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Torsion-free_module&oldid=17214

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098