Let $a$ be a non-zero ideal of an integral doman $A$,whose field of fractions is $K$. Let $a'$ denote the set of $x \in K$, the field of fractions of an integral domain $A$, such that $xa \subset A$. In Local Fields: Pierre-Serre, it says that if $A$ is Noetherian, then $a'$ is a fractional ideal. Isn't $a'$ a fractional ideal regardless of whether $A$ is Noetherian or not?

My attempt: Let $y$ be a non-zero element of $a$, then by definition of $a'$, $ya' \subset A \implies a'$ is a fractional ideal (by the Wikipedia definition of fractional ideal).


The issue is that Serre is taking a different definition of "fractional ideal" than Wikipedia; if you look at page 11, you'll see that to Serre, a fractional ideal is an $A$-submodule of $K$ which is finitely generated over $A$. Serre's definition implies the definition you see on Wikipedia, and while the converse is not true in general, the two definitions are equivalent when $A$ is Noetherian.

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    $\begingroup$ thanks, I also thought that this was the issue, but I wasn't sure $\endgroup$ – P-addict May 8 '19 at 6:43

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