Is the intersection of two Noetherian rings Noetherian?

Question

Is the intersection of two Noetherian rings also Noetherian?

If yes, could you please give me the idea of proof. If not, give me an counterexample.

Answer 1

No!

Let $K\subset L$ be a field extension. It's well known and easily proven that $R=K+XL[X]$ is noetherian if and only if $[L:K]<\infty$.

Let $R_1=\mathbb F_2(Y^2+Y)+X\mathbb F_2(Y)[X]$, and $R_2=\mathbb F_2(Y^3+Y^2)+X\mathbb F_2(Y)[X]$. Then $R_1,R_2$ are noetherian, while $R_1\cap R_2=\mathbb F_2+X\mathbb F_2(Y)[X]$ is not.