# nLab torsion-free module

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

A module over a ring whose underlying abelian group has trivial torsion subgroup is called torsion-free.

## Definition

### Torsion-free $\mathbb{Z}$-modules

In classical mathematics, a torsion-free $\mathbb{Z}$-module or torsion free abelian group $M$ could be defined using a variant of the zero-divisor property characteristic of integral domains: for all $r$ in $\mathbb{Z}$ and $m$ in $M$, if $r m = 0$, then $r = 0$ or $m = 0$, or the contrapositive, if $r \neq 0$ and $m \neq 0$, then $r m \neq 0$.

There is also an equivalent definition: a torsion-free $\mathbb{Z}$-module $M$ or torsion free abelian group is such that right multiplication by $m$ is injective if $m \neq 0$ and left multiplication by $r$ is injective if $r \neq 0$, where “multiplication” refers to the $\mathbb{Z}$-action.

In constructive mathematics, there are multiple inequivalent ways of defining a torsion-free $\mathbb{Z}$-module. One could define a torsion-free module as a module such that for all $r$ in $\mathbb{Z}$ and $m$ in $M$, if $r m = 0$, then $r = 0$ and $m = 0$. The first definition is valid in all modules with decidable equality, and could be defined using coherent logic, but is not valid for $\mathbb{R}$-modules.

If the module has a tight apartness relation, then one could define a torsion-free $\mathbb{Z}$-module as a module such that for all $r$ in $\mathbb{Z}$ and $m$ in $M$, if $r \neq 0$ and $m \# 0$, then $r m \# 0$. This is valid in $\mathbb{R}$, but is no longer capable of being defined in coherent logic. Similarly, one could define a torsion-free $\mathbb{Z}$-module $M$ is such that right multiplication by $m$ is injective if $m \# 0$ and left multiplication by $r$ is injective if $r \neq 0$.

A torsion-free ring is a monoid object in torsion-free $\mathbb{Z}$-modules.

### Torsion-free $R$-modules

In classical mathematics, given a commutative ring $R$, a torsion-free $R$-module is a module $M$ such that for all $r$ in $Can(R)$, where $Can(R)$ is the multiplicative submonoid of cancellative elements in $R$ and $m$ in $M$, if $r m = 0$, then $r = 0$ or $m = 0$. Equivalently, the contrapositive, if $m \neq 0$, then $r m \neq 0$. Some authors require $R$ to be an integral domain, where $Can(R)$ is the monoid of nonzero elements in $R$.

In constructive mathematics, given a ring $R$, there are multiple inequivalent ways of defining a torsion-free $R$-module. One could define a torsion-free module as a module such that for all $r$ in $Can(R)$ and $m$ in $M$, if $r m = 0$, then $m = 0$. The first definition is valid in all modules with decidable equality, and could be defined using coherent logic, but is not valid for $\mathbb{R}$-modules.

If $M$ has a tight apartness relations, then one could define a torsion-free module as a module such that for all $r$ in $Can(R)$ and $m$ in $M$, if $m \# 0$, then $r m \# 0$. This is valid in $\mathbb{R}$-modules, but is no longer capable of being defined in coherent logic.

A torsion-free $R$-algebra is a monoid object in torsion-free $R$-modules.

## Properties

###### Proposition

Every divisible torsion-free $\mathbb{Z}$-module is a rational vector space.

###### Proposition

Every integral domain $R$ is a torsion-free $R$-module.