Direct limits and minimal ring extensions
David E. Dobbs
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA.
Pages 226–245 | Received 04 March 2023, Accepted 08 August 2023, Published 11 December 2023
Abstract
If \{A_i \rightarrow B_i\} is a directed system of minimal (unital) ring extensions (involving associative unital rings that need not be commutative) and the canonical injection A:=\varinjlim_i A_i \to B:= \varinjlim_i B_i is used to view A as a subring of B, then either A=B or A \subset B is a minimal ring extension. The preceding assertion is the case n=1 of a more general result which assumes that there exists an integer n\geq 0 such that for each i, each chain of rings contained between A_i and B_i has length at most n. For commutative rings, an (upward-)directed union of ramified (resp., decomposed) minimal ring extensions A_i \rightarrow B_i for which each (A_i,M_i) is quasi-local, M_j \cap B_i=M_i whenever i \leq j in I, and each transition map A_i \rightarrow A_j is an integral extension produces a minimal ring extension A:=\varinjlim_i A_i \to \varinjlim_i B_i=:B (that is, \cup_i A_i \rightarrow \cup_i B_i) such that if M:=\cup_i M_i, then the minimal ring extension A/M \subset B/M is ramified (resp., decomposed) and A \subset B is a minimal ring extension. Applications involving denumerable (upward-)directed unions of fields whose “steps” are algebraic are given to algebraically closed fields and to perfect closures (in the sense of Bourbaki), by using the \mu-field extensions of Gilbert and Quigley.
Keywords: Unital associative ring, minimal ring extension, field, noncommutative, field extension, λ-field extension,µ-field extension, algebraically closed field, perfect field, perfect closure, direct limit, length of a chain.
MSC numbers: Primary 16B99; Secondary 12F05, 12F15, 13B99.
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