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Fractional ideal

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A subset $Q$ of the field of fractions $K$ of a commutative integral domain $R$ of the form $Q=a^{-1}I$, where $a\in R$, $a\neq0$, and $I$ is an ideal of $R$. In other words, $Q$ is an $R$-submodule of the field $K$ all elements of which permit a common denominator, i.e. there exists an element $a\in R$, $a\neq0$, such that $ax\in R$ for all $x\in Q$. Fractional ideals form a semi-group $\mathfrak A$ with unit element $R$ 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 $\mathfrak A$ are said to be invertible ideals. Each invertible ideal has a finite basis over $R$.

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)
How to Cite This Entry:
Fractional ideal. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Fractional_ideal&oldid=35634
This article was adapted from an original article by L.A. Bokut (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article