View source for 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 element]]s of the semi-group $\mathfrak A$ are said to be invertible ideals. Each invertible ideal has a finite basis over $R$.

====References====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  O. Zariski,   P. Samuel,   "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR>
</table>