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In abstract algebra, the term 'torsion' refers to a number of concepts related to elements of finite order in groups and to the failure of modules to be free.
Let ''G'' be a group. An element ''g'' of ''G'' is called a 'torsion element' if ''g'' has finite order. If all elements of ''G'' are torsion, then ''G'' is called a 'torsion group'. If the only torsion element is the neutral element, then the group ''G'' is called 'torsion-free'.
Let ''M'' be a module over a ring ''R'' without zero divisors. An element ''m'' of ''M'' is called a 'torsion element' if the cyclic submodule of ''M'' generated by ''m'' is not free. Equivalently, ''m'' is torsion if and only if it has a non-zero annihilator in ''R''. If the ring ''R'' is commutative, then the set of all torsion elements forms a submodule of ''M'', called the 'torsion submodule' of ''M'', sometimes denoted T(''M''). The module ''M'' is called a 'torsion module' if T(''M'') = ''M'', and is called 'torsion-free' if T(''M'') = 0.
If the ring ''R'' is non-commutative then the situation is more complicated, and the set of torsion elements need not be a submodule. Nevertheless, it is submodule under a very general assumption that the ring ''R'' satisfies the Ore condition. This covers the case when ''R'' is a Noetherian ring.
Any abelian group may be viewed as a module over the ring 'Z' of integers, and in this case the two notions of torsion coincide.
More generally, let ''R'' be an arbitrary ring and ''S'' ⊂ ''R'' be a multiplicatively closed subset. Then one defines the notion of ''S''-torsion as follows. An element ''m'' of an ''R''-module ''M'' is called an ' ''S''-torsion element' if there exists ''s'' in ''S'' such that ''s'' annihilates ''m'', i.e., ''sm'' = 0. In particular, one can take for ''S'' to be the set of all non-zero divisors of the ring ''R''. In this case, ''S''-torsion is frequently called simply torsion, extending the definition above from the case of domains to general rings.
1. For any ring ''R'', let ''M'' be a free module. Then it follows immediately from the definitions that ''M'' is torsion-free. In particular, any free abelian group is torsion-free.
Any vector space over a field 'K' is torsion-free when viewed as the module over 'K'. By contrast, any finite group is periodic.
2. In the modular group, 'Γ' obtained from the group SL(2,'Z') of two by two integer matrices with unit determinant by factoring out its center, any nontrivial torsion element either has order two and is conjugate to the element ''S'' or has order three and is conjugate to the element ''ST''. In this case, torsion elements do not form a subgroup, for example, ''S'' · ''ST'' = ''T'', which has infinite order.
3. The abelian group 'Q'/'Z', consisting of the rational numbers (mod 1), is periodic, i.e. every element has finite order. Analogously, the module 'K'(''t'')/'K'[''t''] over the ring ''R'' = 'K'[''t''] of polynomials in one variable is pure torsion. Both these examples can be generalized as follows: if ''R'' is a commutative domain and ''Q'' is its field of fractions, then ''Q''/''R'' is a torsion ''R''-module.
Suppose that ''R'' is a (commutative) principal ideal domain and ''M'' is a finitely-generated ''R''-module. Then the structure theorem for finitely generated modules over a principal ideal domain gives a detailed description of the module ''M'' up to isomorphism. In particular, it claims that
:
where ''F'' is a free ''R''-module of finite rank (depeding only on ''M'') and T(''M'') is the torsion submodule of ''M''. As a corollary, any finite-generated torsion-free module over ''R'' is free. This corollary ''does not'' hold for more general commutative domains, even for ''R'' = 'K'[''x'',''y''], the ring of polynomial in two variables.
Assume that ''R'' is a commutative domain and ''M'' is an ''R''-module. Let ''Q'' be the quotient field of the ring ''R''. Then one can consider the ''Q''-module
:
obtained from ''M'' by extension of scalars. Since ''Q'' is a field, a module over ''Q'' is a vector space, possibly, infinite-dimensional. There is a canonical homomorphism of abelian groups from ''M'' to ''M''''Q'', and the kernel of this homomorphism is precisely the torsion submodule T(''M''). More generally, if ''S'' is a multiplicatively closed subset of the ring ''R'', then we may consider localization of the ''R''-module ''M'',
:
which is a module over the local ring ''R''''S''. There is a canonical map from ''M'' to ''M''''S'', whose kernel is precisely the ''S''-torsion submodule of ''M''.
Thus the torsion submodule of ''M'' can be interpreted as the set of the elements that 'vanish in the localization'. The same interpretation continues to hold in the non-commutative setting for rings satisfying the Ore condition.
The concept of torsion plays an important role in homological algebra. If ''M'' and ''N'' are two modules over a commutative ring ''R'' (for example, two abelian groups, when ''R'' = 'Z'), Tor functors yield a family of ''R''-modules Tor''i''(''M'',''N''). Loosely speaking, nontrivial torsion in ''M'' can be detected by the the higher Tor functors (''i'' greater than zero) with an appropriate module ''N'', at least when the ring ''R'' is a domain. The symbol Tor denoting the functors reflects this relation with the algebraic torsion.
★ Torsion (topology)
★ Localization of a module
★ Flat module
★ Universal coefficient theorem
★ Ernst Kunz, "Introduction to Commutative algebra and algebraic geometry", Birkhauser 1985, ISBN 0-8176-3065-1
★
| Contents |
| Definition |
| Examples |
| Case of a principal ideal domain |
| Torsion and localization |
| Torsion in homological algebra |
| See also |
| References |
Definition
Let ''G'' be a group. An element ''g'' of ''G'' is called a 'torsion element' if ''g'' has finite order. If all elements of ''G'' are torsion, then ''G'' is called a 'torsion group'. If the only torsion element is the neutral element, then the group ''G'' is called 'torsion-free'.
Let ''M'' be a module over a ring ''R'' without zero divisors. An element ''m'' of ''M'' is called a 'torsion element' if the cyclic submodule of ''M'' generated by ''m'' is not free. Equivalently, ''m'' is torsion if and only if it has a non-zero annihilator in ''R''. If the ring ''R'' is commutative, then the set of all torsion elements forms a submodule of ''M'', called the 'torsion submodule' of ''M'', sometimes denoted T(''M''). The module ''M'' is called a 'torsion module' if T(''M'') = ''M'', and is called 'torsion-free' if T(''M'') = 0.
If the ring ''R'' is non-commutative then the situation is more complicated, and the set of torsion elements need not be a submodule. Nevertheless, it is submodule under a very general assumption that the ring ''R'' satisfies the Ore condition. This covers the case when ''R'' is a Noetherian ring.
Any abelian group may be viewed as a module over the ring 'Z' of integers, and in this case the two notions of torsion coincide.
More generally, let ''R'' be an arbitrary ring and ''S'' ⊂ ''R'' be a multiplicatively closed subset. Then one defines the notion of ''S''-torsion as follows. An element ''m'' of an ''R''-module ''M'' is called an ' ''S''-torsion element' if there exists ''s'' in ''S'' such that ''s'' annihilates ''m'', i.e., ''sm'' = 0. In particular, one can take for ''S'' to be the set of all non-zero divisors of the ring ''R''. In this case, ''S''-torsion is frequently called simply torsion, extending the definition above from the case of domains to general rings.
Examples
1. For any ring ''R'', let ''M'' be a free module. Then it follows immediately from the definitions that ''M'' is torsion-free. In particular, any free abelian group is torsion-free.
Any vector space over a field 'K' is torsion-free when viewed as the module over 'K'. By contrast, any finite group is periodic.
2. In the modular group, 'Γ' obtained from the group SL(2,'Z') of two by two integer matrices with unit determinant by factoring out its center, any nontrivial torsion element either has order two and is conjugate to the element ''S'' or has order three and is conjugate to the element ''ST''. In this case, torsion elements do not form a subgroup, for example, ''S'' · ''ST'' = ''T'', which has infinite order.
3. The abelian group 'Q'/'Z', consisting of the rational numbers (mod 1), is periodic, i.e. every element has finite order. Analogously, the module 'K'(''t'')/'K'[''t''] over the ring ''R'' = 'K'[''t''] of polynomials in one variable is pure torsion. Both these examples can be generalized as follows: if ''R'' is a commutative domain and ''Q'' is its field of fractions, then ''Q''/''R'' is a torsion ''R''-module.
Case of a principal ideal domain
Suppose that ''R'' is a (commutative) principal ideal domain and ''M'' is a finitely-generated ''R''-module. Then the structure theorem for finitely generated modules over a principal ideal domain gives a detailed description of the module ''M'' up to isomorphism. In particular, it claims that
:
where ''F'' is a free ''R''-module of finite rank (depeding only on ''M'') and T(''M'') is the torsion submodule of ''M''. As a corollary, any finite-generated torsion-free module over ''R'' is free. This corollary ''does not'' hold for more general commutative domains, even for ''R'' = 'K'[''x'',''y''], the ring of polynomial in two variables.
Torsion and localization
Assume that ''R'' is a commutative domain and ''M'' is an ''R''-module. Let ''Q'' be the quotient field of the ring ''R''. Then one can consider the ''Q''-module
:
obtained from ''M'' by extension of scalars. Since ''Q'' is a field, a module over ''Q'' is a vector space, possibly, infinite-dimensional. There is a canonical homomorphism of abelian groups from ''M'' to ''M''''Q'', and the kernel of this homomorphism is precisely the torsion submodule T(''M''). More generally, if ''S'' is a multiplicatively closed subset of the ring ''R'', then we may consider localization of the ''R''-module ''M'',
:
which is a module over the local ring ''R''''S''. There is a canonical map from ''M'' to ''M''''S'', whose kernel is precisely the ''S''-torsion submodule of ''M''.
Thus the torsion submodule of ''M'' can be interpreted as the set of the elements that 'vanish in the localization'. The same interpretation continues to hold in the non-commutative setting for rings satisfying the Ore condition.
Torsion in homological algebra
The concept of torsion plays an important role in homological algebra. If ''M'' and ''N'' are two modules over a commutative ring ''R'' (for example, two abelian groups, when ''R'' = 'Z'), Tor functors yield a family of ''R''-modules Tor''i''(''M'',''N''). Loosely speaking, nontrivial torsion in ''M'' can be detected by the the higher Tor functors (''i'' greater than zero) with an appropriate module ''N'', at least when the ring ''R'' is a domain. The symbol Tor denoting the functors reflects this relation with the algebraic torsion.
See also
★ Torsion (topology)
★ Localization of a module
★ Flat module
★ Universal coefficient theorem
References
★ Ernst Kunz, "Introduction to Commutative algebra and algebraic geometry", Birkhauser 1985, ISBN 0-8176-3065-1
★
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