Fractional ideal - Knowlpedia

Let $R$ be an integral domain with fraction field $K$ . A fractional ideal of $R$ is a nonzero $R$ -submodule $I\subseteq K$ for which some $0\ne c\in R$ satisfies $cI\subseteq R$ .

We use the convention excluding the zero module. If $R$ is Noetherian, these are exactly the nonzero finitely generated $R$ -submodules of $K$ .

Operations

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 $\mathbb Z$ .

Invertibility

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.

References

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