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
Best Answer
Well, no. $K$ is a field, so it has no non-trivial ideals. But $\mathfrak{a}$ is an $\mathcal{O}_K$-submodule of $K$ (this follows from the fact that $c \mathfrak{a}$ is an ideal).