Torsion-free module

In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is ''torsion free'' if its torsion submodule contains only the zero element.

In integral domains the regular elements of the ring are its nonzero elements, so in this case a torsion-free module is one such that zero is the only element annihilated by some non-zero element of the ring. Some authors work only over integral domains and use this condition as the definition of a torsion-free module, but this does not work well over more general rings, for if the ring contains zero-divisors then the only module satisfying this condition is the zero module.

Examples of torsion-free modules

Over a commutative ring ''R'' with total quotient ring ''K'', a module ''M'' is torsion-free if and only if Tor₁(''K''/''R'',''M'') vanishes. Therefore flat modules, and in particular free and projective modules, are torsion-free, but the converse need not be true. An example of a torsion-free module that is not flat is the ideal (''x'', ''y'') of the polynomial ring ''k'' 'x'', ''y''over a field ''k'', interpreted as a module over ''k'' 'x'', ''y''

Any torsionless module over a domain is a torsion-free module, but the converse is not true, as Q is a torsion-free Z-module that is ''not'' torsionless.

Structure of torsion-free modules

Over a Noetherian integral domain, torsion-free modules are the modules whose only associated prime is zero. More generally, over a Noetherian commutative ring the torsion-free modules are those modules all of whose associated primes are contained in the associated primes of the ring.

Over a Noetherian integrally closed domain, any finitely-generated torsion-free module has a free submodule such that the quotient by it is isomorphic to an ideal of the ring.

Over a Dedekind domain, a finitely-generated module is torsion-free if and only if it is projective, but is in general not free. Any such module is isomorphic to the sum of a finitely-generated free module and an ideal, and the class of the ideal is uniquely determined by the module.

Over a principal ideal domain, finitely-generated modules are torsion-free if and only if they are free.

Torsion-free covers

Over an integral domain, every module ''M'' has a torsion-free cover from a torsion-free module ''F'' onto ''M'', with the properties that any other torsion-free module mapping onto ''M'' factors through ''F'', and any endomorphism of ''F'' over ''M'' is an automorphism of ''F''. Such a torsion-free cover of ''M'' is unique up to isomorphism. Torsion-free covers are closely related to flat covers.

Torsion-free quasicoherent sheaves

A quasicoherent sheaf ''F'' over a scheme ''X'' is a sheaf of $\mathcal_X$-modules such that for any open affine subscheme ''U'' = Spec(''R'') the restriction ''F'', _U is associated to some module ''M'' over ''R''. The sheaf ''F'' is said to be torsion-free if all those modules ''M'' are torsion-free over their respective rings. Alternatively, ''F'' is torsion-free if and only if it has no local torsion sections..

See also

* Torsion (algebra)
* torsion-free abelian group
* torsion-free abelian group of rank 1; the classification theory exists for this class.

References

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*{{Citation , author1=The Stacks Project Authors , title=The Stacks Project , url=http://stacks.math.columbia.edu/

Ring theory