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Fractional ideal

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A subset of the field of fractions of a commutative integral domain of the form , where , , and is an ideal of . In other words, is an -submodule of the field all elements of which permit a common denominator, i.e. there exists an element , , such that for all . Fractional ideals form a semi-group with unit element with respect to multiplication. This semi-group is a group for Dedekind rings and only for such rings (cf. Dedekind ring). The invertible elements of the semi-group are said to be invertible ideals. Each invertible ideal has a finite basis over .

References

[1] O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)
[2] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)

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How to Cite This Entry:
Fractional ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fractional_ideal&oldid=31616
This article was adapted from an original article by L.A. Bokut (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article