en>Yobot: /* Examples */Reference before punctuation using AWB (9585)

2013-11-07T15:32:51Z

Examples: Reference before punctuation using AWB (9585)

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	~~I am Oscar~~ and ~~I totally dig~~ that ~~name~~. ~~One~~ of the ~~things she enjoys most~~ is to ~~study comics~~ and ~~she~~'ll be ~~beginning something else alongside~~ with it. ~~Bookkeeping~~ is ~~what I do~~. ~~Puerto Rico~~ is exactly where he and ~~his wife live~~.<br><br>~~Here~~ is ~~my page :~~: [~~http:~~//~~www~~.~~animecontent~~.~~com~~/~~blog~~/~~349119 http~~:/~~/www~~.~~animecontent~~.~~com~~]		{{dablink\|For other notions of torsion, see [[Torsion (disambiguation)\|Torsion]].}}
			In [[abstract algebra]], the term '''torsion''' refers to elements of finite order in [[group (mathematics)\|groups]] and to elements of [[module (mathematics)\|module]]s annihilated by regular elements of a [[ring (mathematics)\|ring]].

			== Definition ==
			An element ''m'' of a [[module (algebra)\|module]] ''M'' over a [[ring (mathematics)\|ring]] ''R'' is called a '''torsion element''' of the module if there exists a [[regular element]] ''r'' of the ring (a non-zero element of the ring that is neither a left nor a right [[zero divisor]]) that annihilates ''m'', i.e., {{nowrap\|1=''r'' ''m'' = 0.}}
			In an [[integral domain]] (a [[commutative ring]] without zero divisors), every non-zero element is regular, so a torsion element of a module over an integral domain is one annihilated by a non-zero element of the integral domain. Some authors use this as the definition of a torsion element but this definition does not work well over more general rings.

			A module ''M'' over a ring ''R'' is called a '''torsion module''' if all its elements are torsion elements, and '''[[Torsion-free module\|torsion-free]]''' if zero is the only torsion element. 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''). If ''R'' is not commutative, T(''M'') may or may not be a submodule. It is shown in {{harv\|Lam\|2007}} that ''R'' is a right [[Ore condition\|Ore ring]] if and only if T(''M'') is a submodule of ''M'' for all right ''R'' modules. Since right Noetherian domains are Ore, this covers the case when ''R'' is a right [[Noetherian ring\|Noetherian]] [[domain (ring theory)\|domain]] (which might not be commutative).

			More generally, let ''M'' be a module over a ring ''R'' and ''S'' be a multiplicatively closed subset of ''R''. An element ''m'' of ''M'' is called an ''' ''S''-torsion element''' if there exists an element ''s'' in ''S'' such that ''s'' annihilates ''m'', i.e., {{nowrap\|1=''s'' ''m'' = 0.}} In particular, one can take for ''S'' the set of regular elements of the ring ''R'' and recover the definition above.

			An element ''g'' of a [[group (mathematics)\|group]] ''G'' is called a '''torsion element''' of the group if it has finite [[order (group theory)\|order]], i.e., if there is a positive [[integer]] ''m'' such that ''g''<sup>''m''</sup> = ''e'', where ''e'' denotes the [[identity element]] of the group, and ''g''<sup>''m''</sup> denotes the product of ''m'' copies of ''g''. A group is called a '''[[torsion group \|torsion (or periodic) group]]''' if all its elements are torsion elements, and a '''torsion-free group''' if the only torsion element is the identity element. Any [[abelian group]] may be viewed as a module over the ring '''Z''' of [[integer]]s, and in this case the two notions of torsion coincide.

			== Examples ==

			# Let ''M'' be a [[free module]] over any ring ''R''. Then it follows immediately from the definitions that ''M'' is torsion-free (if the ring ''R'' is not a domain then torsion is considered with respect to the set ''S'' of non-zero divisors of ''R''). In particular, any [[free abelian group]] is torsion-free and any [[vector space]] over a field '''K''' is torsion-free when viewed as the module over '''K'''.
			# By contrast with Example 1, any [[finite group]] (abelian or not) is periodic and finitely generated. [[Burnside's problem]] asks whether, conversely, any finitely generated periodic group must be finite. (The answer is "no" in general, even if the period is fixed.)
			# 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.
			# 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 [[polynomial]]s 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.
			# The torsion subgroup of ('''R'''/'''Z''',+) is ('''Q'''/'''Z''',+) while the groups ('''R''',+),('''Z''',+) are torsion-free. The quotient of a [[torsion-free abelian group]] by a subgroup is torsion-free exactly when the subgroup is a [[pure subgroup]].
			# Consider a linear operator '''L''' acting on a finite-dimensional vector space '''V'''. If we view '''V''' as an '''F'''['''L''']-module in the natural way, then (as a result of many things, either simply by finite-dimensionality or as a consequence of the [[Cayley–Hamilton theorem]]), '''V''' is a torsion '''F'''['''L''']-module.

			== Case of a principal ideal domain ==

			Suppose that ''R'' is a (commutative) [[principal ideal domain]] and ''M'' is a [[finitely-generated module\|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

			: <math> M \simeq F\oplus T(M), </math>

			where ''F'' is a free ''R''-module of finite rank (depending only on ''M'') and T(''M'') is the torsion submodule of ''M''. As a corollary, any finitely-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 polynomials in two variables.
			For non-finitely generated modules, the above direct decomposition is not true. The torsion subgroup of an abelian group may not be a direct summand of it.

			== 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

			: <math> M_Q = M \otimes_R Q, </math>

			obtained from ''M'' by [[extension of scalars]]. Since ''Q'' is a [[field (mathematics)\|field]], a module over ''Q'' is a [[vector space]], possibly, infinite-dimensional. There is a canonical homomorphism of abelian groups from ''M'' to ''M''<sub>''Q''</sub>, and the [[Kernel (algebra)\|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 a module\|localization]] of the ''R''-module ''M'',

			: <math> M_S = M \otimes_R R_S, </math>

			which is a module over the [[localization of a ring\|localization]] ''R''<sub>''S''</sub>. There is a canonical map from ''M'' to ''M''<sub>''S''</sub>, 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, or more generally for any [[Ore condition#Multiplicative sets\|right denominator set]] ''S'' and right ''R''-module ''M''.

			== 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 functor]]s yield a family of ''R''-modules Tor<sub>''i''</sub>(''M'',''N''). The ''S''-torsion of an ''R''-module ''M'' is canonically isomorphic to Tor<sub>1</sub>(''M'', ''R''<sub>''S''</sub>/''R''). The symbol Tor denoting the functors reflects this relation with the algebraic torsion. This same result holds for non-commutative rings as well as long as the set ''S'' is a [[Ore condition#Multiplicative sets\|right denominator set]].

			== Abelian varieties ==
			[[Image:Lattice torsion points.svg\|right\|thumb\|300px\|The 4-torsion subgroup of an elliptic curve over the complex numbers.]]
			The torsion elements of an [[abelian variety]] are ''torsion points'' or, in an older terminology, ''division points''. On [[elliptic curve]]s they may be computed in terms of [[division polynomials]].

			==See also==
			* [[Analytic torsion]]
			* [[Arithmetic dynamics]]
			* [[Flat module]]
			* [[Localization of a module]]
			* [[Rank of an abelian group]]
			* [[Ray-Singer torsion]]
			* [[Torsion-free abelian group]]
			* [[Universal coefficient theorem]]

			==References==

			*Ernst Kunz, "Introduction to Commutative algebra and algebraic geometry", Birkhauser 1985, ISBN 0-8176-3065-1
			*Irving Kaplansky, "Infinite abelian groups", University of Michigan, 1954.
			*{{springer\|id=T/t093330\|title=Torsion submodule\|author=[[Michiel Hazewinkel]]}}
			*{{citation \|last=Lam\|first=T. Y. \|title=Exercises in modules and rings \|series=Problem Books in Mathematics \|publisher=Springer \|place=New York \|year=2007 \|pages=xviii+412 \|isbn=978-0-387-98850-4 \|isbn=0-387-98850-5 \|mr=2278849 \|doi=10.1007/978-0-387-48899-8}}

			{{DEFAULTSORT:Torsion (Algebra)}}
			[[Category:Abelian group theory]]
			[[Category:Module theory]]
			[[Category:Homological algebra]]

en>Cydebot: Robot - Speedily moving category Relativity to :Category:Theory of relativity per CFDS.

2012-03-23T18:41:41Z

Robot - Speedily moving category Relativity to Category:Theory of relativity per CFDS.

New page

I am Oscar and I totally dig that name. One of the things she enjoys most is to study comics and she'll be beginning something else alongside with it. Bookkeeping is what I do. Puerto Rico is exactly where he and his wife live.<br><br>Here is my page :: [http://www.animecontent.com/blog/349119 http://www.animecontent.com]

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en>Yobot: /* Examples */Reference before punctuation using AWB (9585)

en>Cydebot: Robot - Speedily moving category Relativity to :Category:Theory of relativity per CFDS.