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In abstract algebra, the term torsion refers to elements of finite order in groups and to elements of modules annihilated by regular elements of a ring. == Definition == An element ''m'' of a module ''M'' over a ring ''R'' is called a torsion element of the module if there exists a regular element ''r'' of the ring (an element that is neither a left nor a right zero divisor) that annihilates ''m'', i.e., 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 if zero is the only torsion element. If the ring ''R'' is an integral domain 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 that ''R'' is a right 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 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., 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 ''G'' is called a torsion element of the group if it has finite order, i.e., if there is a positive integer ''m'' such that ''g''''m'' = ''e'', where ''e'' denotes the identity element of the group, and ''g''''m'' denotes the product of ''m'' copies of ''g''. A group is called a 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 integers, and in this case the two notions of torsion coincide. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Torsion (algebra)」の詳細全文を読む スポンサード リンク
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