symmetric monoidal (∞,1)-category of spectra
A module over a ring whose underlying abelian group has trivial torsion subgroup is called torsion-free.
In classical mathematics, a torsion-free -module or torsion free abelian group could be defined using a variant of the zero-divisor property characteristic of integral domains: for all in and in , if , then or , or the contrapositive, if and , then .
There is also an equivalent definition: a torsion-free -module or torsion free abelian group is such that right multiplication by is injective if and left multiplication by is injective if , where “multiplication” refers to the -action.
In constructive mathematics, there are multiple inequivalent ways of defining a torsion-free -module. One could define a torsion-free module as a module such that for all in and in , if , then and . The first definition is valid in all modules with decidable equality, and could be defined using coherent logic, but is not valid for -modules.
If the module has a tight apartness relation, then one could define a torsion-free -module as a module such that for all in and in , if and , then . This is valid in , but is no longer capable of being defined in coherent logic. Similarly, one could define a torsion-free -module is such that right multiplication by is injective if and left multiplication by is injective if .
A torsion-free ring is a monoid object in torsion-free -modules.
In classical mathematics, given a commutative ring , a torsion-free -module is a module such that for all in , where is the multiplicative submonoid of cancellative elements in and in , if , then or . Equivalently, the contrapositive, if , then . Some authors require to be an integral domain, where is the monoid of nonzero elements in .
In constructive mathematics, given a ring , there are multiple inequivalent ways of defining a torsion-free -module. One could define a torsion-free module as a module such that for all in and in , if , then . The first definition is valid in all modules with decidable equality, and could be defined using coherent logic, but is not valid for -modules.
If has a tight apartness relations, then one could define a torsion-free module as a module such that for all in and in , if , then . This is valid in -modules, but is no longer capable of being defined in coherent logic.
A torsion-free -algebra is a monoid object in torsion-free -modules.
Every divisible torsion-free -module is a rational vector space.
Every integral domain is a torsion-free -module.
free module projective module flat module torsion-free module
See also
Last revised on June 17, 2022 at 20:49:18. See the history of this page for a list of all contributions to it.