I'm confused with the definition of fractional ideals.
In "Advanced Modern Algebra by Rotman" the definition is as follows: If $R$ is a domain with $F$ = Frac$(R)$, then a fractional ideal is a finitely generated nonzero $R$-submodule of $F$.
However, in other sources they define it as: Let $R$ be an integral domain, and let $F$ be its field of fractions. A fractional ideal of R is an $R$-submodule $I$ of $F$ such that there exists a non-zero $r\in R$ such that $rI \subset R$.
So in the second definition, a fractional ideal is not necessarily finitely generated module. For example, if $R$ is noetherian then it is finitely generated.
Which one is correct?