Let $K$ be the quotient field of an integral domain $R$. A fractional ideal $I$ is a subset of $K$ not $\{0\}$, for which a $0 \neq r \in R$ exists so that $r I \subseteq R$ is an ideal in $R$.
All examples I've found are either very general or complicated. I'm looking for simple examples of fractional ideals.
Wouldn't just be $$I = \biggl\{\ldots, -\frac{15}{7}, -\frac{10}{7}, -\frac{5}{7}, 0, \frac{5}{7}, \frac{10}{7}, \frac{15}{7}, \ldots \biggr\}$$ an example of a fractional ideal with $R=\mathbb{Z}$ and $K = \mathbb{Q}$ (because $7 \cdot I = 5 \mathbb{Z}$)?
Thanks in advance!