Kyungpook Mathematical Journal 2020; 60(1): 53-69

**Published online** March 31, 2020

Copyright © Kyungpook Mathematical Journal.

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

David Earl Dobbs

Department of Mathematics, University of Tennessee, Knoxville, Tennessee 379961320, USA

e-mail : dedobbs@comporium.net

**Received**: June 12, 2019; **Revised**: January 7, 2020; **Accepted**: January 10, 2020

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 field.

**Keywords**: commutative ring, ring extension, minimal ring extension, inert extension, maximal ideal, minimal ﬁ,eld extension, quasi-local ring, integrality, pullback.