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I'm going through some notes, and have the following definition:

Let $K$ be a number field. Then $ \mathfrak{a} \subset K$ is a fractional ideal if there exists a non-zero $c \in K$ such that $c\mathfrak{a} \subset \mathcal O_K$ is an ideal.

I'm concerned that this is unclearly stated; specifically, shouldn't it specify that $\mathfrak{a}$ is an ideal of $K$? If $\mathfrak{a}$ is just any subset of $K$, then I can't prove the lemma that gives the correspondence between fractional ideals and finitely generated $\mathcal O_K$ modules.

Thanks

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    An $R$-submodule of a ring $R$ is an ideal in $R$2012-02-28
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    Just to be concrete: Let $K=\mathbb{Q}$. Then $\mathcal{O}_K=\mathbb{Z}$. Then the $\mathbb{Z}$-module (=abelian group) generated by (1/2) is a fractional ideal, because if you multiply it by 2, you get an ideal.2012-02-28

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