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eISSN 0454-8124
pISSN 1225-6951

Article

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

Published online March 31, 2020

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

Abstract

Let (A,M) ⊂ (B,N) be commutative quasi-local rings. We consider the property that there exists a ring D such that ADB and the extension DB 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 MN. Suppose henceforth that MN. If the field extension A/MB/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/MB/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.