Let $R$ be a Noetherian domain with quotient field $K$ and let $b_1,\ldots,b_n\in K$.
Suppose that $R'$ is a integral domain, $R\subseteq R'$ and
$$R'\subseteq \sum_j Rb_j.$$
Remark: It is well know that if $R$ is a Noetherian ring and $M$ is a finitely generated $R$-module then $M$ is Noetherian.
Thus, the $R$-module $\sum_j Rb_j$ is Noetherian.
Now, let $I$ be a ideal of $R'$, then $I$ is a $R$-submodule of $\sum_j Rb_j.$
This implies that $I$ is a finitely generated $R$-submodule, in particular is finitely generated as $R'$-module.
The conclusion is that $R'$ is a Noetherian ring.
Is correct ?
Thank you all.