Recent Progress on Minimal Ring Extensions and Related Concepts

摘要

A unital extension A is contained in B of (commutative unital) rings is said to be a minimal ring extension if there is no ring C such that A is contained in C is contained in B. The introductory section summarizes the background needed to study this concept. Section 2 describes several themes pertaining to the classification of minimal ring extensions, including the classification in 2006 by Dobbs-Shapiro of the minimal ring extensions of an arbitrary (integral) domain and various recent generalizations of the fact that any minimal domain extension of a non-field must be an overring. Section 3 describes some ways that minimal ring extensions arise naturally. These include chain-theoretic studies, such as the recent result by Coykendall-Dobbs that, despite the known case for finite chains, a domain R with a saturated chain consisting of integrally closed overrings need not be a Prufer domain; and FIP-theoretic studies generalizing the Primitive Element Theorem of field theory. Among the FlP-theoretic results are the generalization in 2003 by Dobbs-Mullins-Picavet-Picavet-L'Hermitte of the Primitive Element Theorem by characterizing, for each field K, the commutative unital K-algebras that have only finitely many unital K-subalgebras; and the recent characterization by Dobbs-Picavet-Picavet-L'Hermitte of the rings having only finitely many unital subrings. Also surveyed in regard to the latter topic are recent results concerning whether composites of minimal ring extensions (resp., ring extensions satisfying FIP) R is contained in S and R is contained in T are such that S is contained in ST is a minimal ring extension (resp., satisfies FIP). The final section summarizes recent results that use variants of the classical Kaplansky transform to characterize minimal ring extensions R is contained in T in which R is integrally closed in T; and, to illustrate similarities and differences between allied concepts, quotes several results from an unpublished dissertation of M. S. Gilbert on the generalization of minimal ring extensions that concerns ring extensions whose set of intermediate rings is linearly ordered by inclusion.