In the wikipedia article http://en.wikipedia.org/wiki/Fractional_ideal we read
Let $R $ be an integral domain, and let $K$ be its field of fractions. A fractional ideal of $R$ is an $R$-submodule $I$ of $K$ such that there exists a non-zero $r\in R$ such that $rI\subset R$.
1) $I$ is an $R$-submodule of what $R$-module? my guess: $K$ being the field of fractions of $R$, then an element $r\in R$ can be seen as ${r\over 1}\in K$ so we can define the scalar mulitplication $r*{r_1\over r_2}={r\over 1}*{r_1\over r_2}={rr_1\over r_2}$ hence $K$ is an $R$-module and $I$ is an $R$-submodule of $K$.
2) Why do we call it a fractional ideal of the field $K$ knowing that a field does not have any proper ideal?
3)if $R=\mathbb Z$ then $K=\mathbb Q$ what are the fractional ideals of $\mathbb Q$?