# Torsion-free module

A module $M$ over a ring $A$ without divisors of zero, such that the equality $am=0$, where $a\in A$, $m\in M$, implies $a=0$ or $m=0$. Examples of such (left) modules are the ring $A$ 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 $A$ is commutative, then for any module $M$ there is a torsion submodule

$$T(M)=\{m\in M\colon\exists a\in A,a\neq0,am=0\}.$$

In this case the quotient module $M/T(M)$ is torsion-free.

#### Comments

More generally, for any associative ring $R$ a left $R$-module $M$ is called torsion-free if for $m\in M$, $rm=0$ for a regular element $r\in R$ implies $m=0$. Cf. Torsion submodule for more details and some references.

**How to Cite This Entry:**

Torsion-free module.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Torsion-free_module&oldid=33072