Let $B$ be a DVR, and $A$ is a subring of $B$ such that $\{0,1\}\subseteq A$, we know in general $A$ is not a DVR for the valuation restricted by the valuation of $B$, for example see the answer of this question. But at least I want to know if $A$ is a Noetherian ring?
Thanks!
If you don't assume any relation between the valuation and the subring, you can't deduce anything.
For instance, let $K=k(X_1,X_2,\dots)$ be the field of rational fractions in countably infinite variables. Then $A=K[[Y]]$ is a DVR, and $B=k[X_1,X_2,\dots]$ is a subring of $A$ which is not noetherian.
More generally, if $B$ is any integral ring, let $K$ be the fraction field of $B$. Then $B$ is a subring of the DVR $A=K[[Y]]$. So there is absolutely nothing you can say about a subring of a DVR except it is a domain.