A note on finite dimensional subrings of polynomial rings

Abstract

Let k be a field and {Xx}xe a a family of indeterminates over k.We show that if A is a ring of Knill dimension d such that k c A c. k[{Ax}xe^] then there are elements Yt,''■' •, Y* which are algebraically independent over k and a fc-isomorphism <f> such that k c. (f>(A) c k[Ylt-■ ■ , Yd].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.