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In algebra, a torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring. 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 0 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''() over a field ''k''. * Any torsionless module is a torsion-free module, but the converse is not true, as Q is a torsion-free Z-module which is ''not'' torsionless. 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-free module」の詳細全文を読む スポンサード リンク
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