Fractional ideal

Equivalently, $I=c^{-1}J$ for a nonzero ordinary ideal $J\subseteq R$ and $0\ne c\in R$: take $J=cI$. An ordinary ideal is sometimes called an integral ideal to emphasize containment in $R$.

If $R$ is Noetherian, fractional ideals are exactly the nonzero finitely generated $R$-submodules of $K$. Finite generators have a common denominator; conversely, $cI$ is a finitely generated ideal and multiplication by $c$ identifies it with $I$.

For example, $\tfrac12\mathbb Z$ is fractional, while $\mathbb Z[1/2]\subset\mathbb Q$ is not: no single nonzero integer clears the denominators $2^m$ for every $m$.

The product $IJ$ consists of finite sums of products $xy$, with $x\in I$, $y\in J$. A principal fractional ideal has the form $aR$, $a\in K^\times$. It need not lie inside $R$; for example $\tfrac12\mathbb Z$ is a fractional ideal of $\mathbb Z$.

A fractional ideal is invertible when it has a multiplicative inverse. Over a Dedekind domain, each fractional ideal has inverse $I^{-1}=\{x\in K:xI\subseteq R\}$, with $II^{-1}=R$. This assertion is not valid for every integral domain; nonmaximal orders supply examples.

The multiplier ring $(I:I)$ and colon quotient $(R:I)$ record two different scalar-preservation conditions.

1. J. S. Milne, Algebraic Number Theory, v3.08. Author’s text, §3, “The ideal class group,” Theorem 3.20.