If $B$ is a Noetherian ring and $A$ is a subring of $B$, is $B$ Noetherian as an $A$ module?

Question

Let $A\subset B$ and both $A$,$B$ are ring. Suppose given that $B$ is a noetherian ring then is it true that when we consider $B$ as $A$-module then it is again a noetherian $A$-module?

My effort : I think it is not true as such but unable to find an counter example.Any help/hint in this regards would be highly appreciated. Thanks in advance!

Answer 1

Consider $\mathbb Q\subseteq \mathbb R$.