# Noetherian integral domain and fractional ideal

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.