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).