Does Noetherian + finite dimenasional imply essentially of finite type?

Question

Let $R$ be a Noetherian (commutative) algebra over a field $k$.

If $\dim R<\infty$ (Krull dimension), does it follow that $R$ is essentially of finite type over $k$? (Meaning: $R=S^{-1}(k[x_1,\ldots,x_n]/I)$ is the localization with respect to a multiplicative system $S$ of a $k$-algebra of finite type)

Answer 1

No. For instance, $R$ could be a field extension of $k$ of infinite transcendence degree.