66:
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,
219:, 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.
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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.
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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
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128:, are torsion-free, but the converse need not be true. An example of a torsion-free module that is not flat is the
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204:, any finitely-generated torsion-free module has a free submodule such that the
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is unique up to isomorphism. Torsion-free covers are closely related to
256:, with the properties that any other torsion-free module mapping onto
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is torsion-free if and only if it has no local torsion sections.
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integral domain, torsion-free modules are the modules whose only
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are torsion-free over their respective rings. Alternatively,
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is a torsion-free module, but the converse is not true, as
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481:, The University of Chicago Press, Chicago-London,
421:; the classification theory exists for this class.
341:
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51:) of the ring. In other words, a module is
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234:Over an integral domain, every module
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419:torsion-free abelian group of rank 1
39:such that zero is the only element
342:{\displaystyle {\mathcal {O}}_{X}}
296:Torsion-free quasicoherent sheaves
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185:Structure of torsion-free modules
349:-modules such that for any open
75:Examples of torsion-free modules
59:contains only the zero element.
284:. Such a torsion-free cover of
154:, interpreted as a module over
97:is torsion-free if and only if
1:
493:The Stacks Project Authors,
460:Encyclopedia of Mathematics
248:from a torsion-free module
529:
414:torsion-free abelian group
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238:has a torsion-free cover
212:to an ideal of the ring.
441:Stacks Project, Tag 0AVQ
202:integrally closed domain
16:Not to be confused with
116:) vanishes. Therefore
343:
224:principal ideal domain
455:"Torsion-free_module"
391:if all those modules
344:
478:Torsion-free modules
322:
120:, and in particular
302:quasicoherent sheaf
230:Torsion-free covers
88:total quotient ring
29:torsion-free module
496:The Stacks Project
339:
200:Over a Noetherian
173:is a torsion-free
163:torsionless module
126:projective modules
18:Torsionless module
409:Torsion (algebra)
57:torsion submodule
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351:affine subscheme
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260:factors through
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195:associated prime
177:-module that is
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64:integral domains
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375:to some module
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217:Dedekind domain
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383:. The sheaf
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278:automorphism
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53:torsion free
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49:zero-divisor
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513:Ring theory
362:restriction
290:flat covers
93:, a module
69:zero module
41:annihilated
426:References
373:associated
264:, and any
210:isomorphic
191:Noetherian
465:EMS Press
208:by it is
140:) of the
507:Category
475:(1972),
403:See also
206:quotient
487:0344237
467:, 2001
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