nLab torsion-free module

Contents

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 MM could be defined using a variant of the zero-divisor property characteristic of integral domains: for all rr in \mathbb{Z} and mm in MM, if rm=0r m = 0, then r=0r = 0 or m=0m = 0, or the contrapositive, if r0r \neq 0 and m0m \neq 0, then rm0r m \neq 0.

There is also an equivalent definition: a torsion-free \mathbb{Z}-module MM or torsion free abelian group is such that right multiplication by mm is injective if m0m \neq 0 and left multiplication by rr is injective if r0r \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 rr in \mathbb{Z} and mm in MM, if rm=0r m = 0, then r=0r = 0 and m=0m = 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 rr in \mathbb{Z} and mm in MM, if r0r \neq 0 and m#0m \# 0, then rm#0r 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 MM is such that right multiplication by mm is injective if m#0m \# 0 and left multiplication by rr is injective if r0r \neq 0.

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

Torsion-free RR-modules

In classical mathematics, given a commutative ring RR, a torsion-free RR-module is a module MM such that for all rr in Can(R)Can(R), where Can(R)Can(R) is the multiplicative submonoid of cancellative elements in RR and mm in MM, if rm=0r m = 0, then r=0r = 0 or m=0m = 0. Equivalently, the contrapositive, if m0m \neq 0, then rm0r m \neq 0. Some authors require RR to be an integral domain, where Can(R)Can(R) is the monoid of nonzero elements in RR.

In constructive mathematics, given a ring RR, there are multiple inequivalent ways of defining a torsion-free RR-module. One could define a torsion-free module as a module such that for all rr in Can(R)Can(R) and mm in MM, if rm=0r m = 0, then m=0m = 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 MM has a tight apartness relations, then one could define a torsion-free module as a module such that for all rr in Can(R)Can(R) and mm in MM, if m#0m \# 0, then rm#0r m \# 0. This is valid in \mathbb{R}-modules, but is no longer capable of being defined in coherent logic.

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

Properties

Proposition

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

Proposition

Every integral domain RR is a torsion-free RR-module.

References

See also

Last revised on June 17, 2022 at 16:49:18. See the history of this page for a list of all contributions to it.