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In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed ''integral ideals'' for clarity. ==Definition and basic results== Let ''R'' be an integral domain, and let ''K'' be its field of fractions. A fractional ideal of ''R'' is an ''R''-submodule ''I'' of ''K'' such that there exists a non-zero ''r'' ∈ ''R'' such that ''rI'' ⊆ ''R''. The element ''r'' can be thought of as clearing out the denominators in ''I''. The principal fractional ideals are those ''R''-submodules of ''K'' generated by a single nonzero element of ''K''. A fractional ideal ''I'' is contained in ''R'' if, and only if, it is an ('integral') ideal of ''R''. A fractional ideal ''I'' is called invertible if there is another fractional ideal ''J'' such that ''IJ'' = ''R'' (where ''IJ'' = is called the product of the two fractional ideals). In this case, the fractional ideal ''J'' is uniquely determined and equal to the generalized ideal quotient : The set of invertible fractional ideals form an abelian group with respect to above product, where the identity is the unit ideal ''R'' itself. This group is called the group of fractional ideals of ''R''. The principal fractional ideals form a subgroup. A (nonzero) fractional ideal is invertible if, and only if, it is projective as an ''R''-module. Every finitely generated ''R''-submodule of ''K'' is a fractional ideal and if ''R'' is noetherian these are all the fractional ideals of ''R''. 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「Fractional ideal」の詳細全文を読む スポンサード リンク
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