On residual variables and stably polynomial algebras

Abstract

Abstract Let R be a commutative noetherian ring and let R[X,Y] be a polynomial algebra in two variables over R. Let W be an element of R[X,Y]. In this paper we show that R[X,Y] is a stably polynomial algebra over R[W] if and only if W is a residual variable in R[X,Y]. Moreover, in this case, if either R contains the field of rationals or if Rred is seminormal, then W is a variable in R[X,Y].