$R\subset A\subset R$, $A$ is Noetherian. Is $R$ Noetherian?

Let $R\subset A\subset R[X]$ be commutative rings and suppose $A$ is Noetherian. Is $R$ Noetherian?

I guess the answer is yes. Can we say from this relation that $A[X]=R[X]$? If yes, then by Hilbert Basis Theorem, $A[X]$ is Noetherian, hence $R[X]$ is Noetherian $\Longrightarrow R$ is Noetherian.

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