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Flat module
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=== Non-examples === * If {{mvar|I}} is an ideal in a Noetherian commutative ring {{mvar|R}}, then <math>R/I</math> is not a flat module, except if {{mvar|I}} is generated by an [[idempotent]] (that is an element equal to its square). In particular, if {{mvar|R}} is an [[integral domain]], <math>R/I</math> is flat only if <math>I</math> equals {{mvar|R}} or is the [[zero ideal]]. * Over an integral domain, a flat module is [[torsion-free module|torsion free]]. Thus a module that contains nonzero torsion elements is not flat. In particular <math>\Q/\Z</math> and all fields of positive characteristics are non-flat <math>\Z</math>-modules, where <math>\Z</math> is the ring of integers, and <math>\Q</math> is the field of the rational numbers.
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