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http://dx.doi.org/10.5666/KMJ.2020.60.1.53

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

Dobbs, David Earl (Department of Mathematics, University of Tennessee)
Publication Information
Kyungpook Mathematical Journal / v.60, no.1, 2020 , pp. 53-69 More about this Journal
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 field.
Keywords
commutative ring; ring extension; minimal ring extension; inert extension; maximal ideal; minimal field extension; quasi-local ring; integrality; pullback;
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