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 : firstname.lastname@example.org
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.