Universally Coefficient Domains and Their Relation to the Cancellation Problem for Rings Original Research Article

We call an integral domainDauniversally coefficient domainif for any domainRwithDsubset of or equal toR[x1,…, xn] thenDsubset of or equal toR. It is true that every universally coefficient domain is strongly invariant but not conversely. We show that ifKis a field of characteristic zero andD=S−1K[x, y] thenDis a universally coefficient domain if and only if[formula], wherepis such that[formula]for some[formula]and where[formula]is the algebraic closure ofK. We then prove that any localization ofK[x, y], whereKis algebraically closed of characteristic zero, is a universally coefficient domain if and only if it is strongly invariant, giving necessary and sufficient conditions for localizations ofK[x, y] to be strongly invariant. This, in turn, shows that every localization ofK[x, y],Kalgebraically closed of characteristic zero, is invariant and strongly invariant if it is not a polynomial ring. We also discuss generalizations tonvariables and overrings, and we give examples of classes of polynomialsfsuch thatK[x1,…, xn, 1/f] is a universally coefficient domain.