Fractional ideal

Remarks

Example: $I = \tfrac12 \Z \subseteq \Q$ for $R = \Z$; it is a nonzero $\Z$-submodule, and $2I = \Z \subseteq \Z$, so it is a fractional ideal, with inverse $2\Z$. Non-example: $\Z[\tfrac12] = \set{a/2^k : a \in \Z, k \ge 0}$ is a nonzero $\Z$-submodule of $\Q$, but it fails clause 3: for any nonzero integer $r$, choose $k$ with $2^k > \abs{r}$; then $r/2^k \in r\Z[\tfrac12]$ is not an integer. Every nonzero ideal of $R$ is a fractional ideal, with $r = 1$; these are the integral ideals. For $x \in K^{\times}$, the set $xR$ is a principal fractional ideal, with inverse $x^{-1}R$. When $R$ is Noetherian, clause 3 is equivalent to $I$ being a finitely generated $R$-module. Some authors, and Mathlib, also count $\set{0}$ as a fractional ideal.