A note on finite dimensional subrings of polynomial rings

Abstract:

Let $k$ be a field and ${\{ {X_\alpha }\} _{\alpha \in \Delta }}$ a family of indeterminates over $k$. We show that if $A$ is a ring of Krull dimension $d$ such that $k \subseteq A \subseteq k[{\{ {A_\alpha }\} _{\alpha \in \Delta }}]$ then there are elements ${Y_1}, \cdots ,{Y_d}$ which are algebraically independent over $k$ and a $k$-isomorphism $\phi$ such that $k \subseteq \phi (A) \subseteq k[{Y_1}, \cdots ,{Y_d}]$. This is used to show that a onedimensional ring $A$ which satisfies the above conditions is necessarily an affine ring over $k$ and is necessarily a polynomial ring if it is normal. In addition we show that such a ring $A$ is a normal affine ring of transcendence degree two over $k$ if and only if it is a two-dimensional Krull ring such that each essential valuation of $A$ has residue field transcendental over $k$.