Universally coefficient domains and their relation to the cancellation problem for rings (Q1911395)

An integral domain \(D\) is said to be a universally coefficient domain if whenever \(D\) is contained in a polynomial ring \(R[X_1, \dots, X_n]\), where \(R\) is a domain, it holds that \(D \subseteq R\). The ring \(D\) is said to be invariant if \(D[X_1, \dots, X_n]\cong R[Y_1, \dots, Y_n]\), where the \(X_i\) and \(Y_i\) are indeterminates, implies \(D \cong R\), and \(D\) is said to be strongly invariant if any isomorphism of rings \(\varphi : D[X_1, \dots, X_n]\to R[Y_1, \dots, Y_n]\) satisfies \(\varphi (D) = R\). It is known that universally coefficient domains are strongly invariant, but not conversely. This paper contains many results on these properties. The main ones are the following: Theorem 3.6. Let \(K\) be a field of characteristic zero, \(S\) a multiplicative subset of \(K[X,Y]\) and \(D = S^{-1} K[X,Y]\). Then \(D\) is a universally coefficient domain if and only if \(S \subseteq \overline K[p]\), where \(\overline K\) is the algebraic closure of \(K\), and \(p\) is such that \(\overline K [X,Y] = \overline K[p,q]\) for some \(q \in \overline K[X,Y]\). Theorem 3.7. Let \(K\) be an algebraically closed field of characteristic zero, and let \(D\) be a localization of \(K[X,Y]\). Then \(D\) is strongly invariant if and only if it is not a polynomial ring.