Where Some Inert Minimal Ring Extensions of a Commutative Ring Come from

Abstract

Let (A,M) ⊂ (B,N) be commutative quasi-local rings. We consider the property that there exists a ring D such that A ⊆ D ⊂ B and the extension D ⊂ B is inert. Examples show that the number of such D may be any non-negative integer or infinite. The existence of such D does not imply M ⊆ N . Suppose henceforth that M ⊆ N . If the field extension A/M ⊆ B/N is algebraic, the existence of such D does not imply that B is integral over A (except when B has Krull dimension 0). If A/M ⊆ B/N is a minimal field extension, there exists a unique such D, necessarily given by D = A+N (but it need not be the case that N = MB). The converse fails, even if M = N and B/M is a finite