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In the wikipedia article 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 $\mathbb Q$?

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up vote 5 down vote accepted

Question 1) Yes. The definition says $I$ is an $R$-submodule of $K$. And also, yes, the action is simply "scalar" multiplication. More generally, if $R$ is a subring of any ring $S$, then $S$ is an $R$-module via the action $r\cdot s=rs$ (for $r\in R$ and $s\in S$).

Question 2) $R$-submodules of $R$ are ideals so if $R$ is a field it won't have any interesting submodules of itself. But you are considering $R$-submodules of $K$. These no longer need to be ideals (of $K$) but simply subgroups under addition which are closed under "scalar" multiplication by $R$ (which is considerably weaker than being an ideal). Notice it is a fractional ideal of $R$ (not $K$). Fractional ideals do not have to be subsets of $R$ (which at first seems weird, because ideals are subsets + other conditions).

Question 3) Let $I$ be a fractional ideal of $\mathbb{Z}$. Then $I$ is a $\mathbb{Z}$-submodule of $\mathbb{Q}$ (i.e. a subgroup of $\mathbb{Q}$ under addition) and there exists some $r \in \mathbb{Z}$ such that $rI \subset \mathbb{Z}$. This implies $rI = m\mathbb{Z}$ for some $m \in \mathbb{Z}$. So $I = \frac{m}{r}\mathbb{Z}$ (multiples of the fraction $m/r$). Thus the name "fractional ideal". The ideals of $\mathbb{Z}$ are $J=x\mathbb{Z}$ for $x\in\mathbb{Z}$ whereas the fractional ideals allow $x$ to be a fraction.

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