On the category of algebras over a finite direct product of
commutative rings
David E. Dobbs
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA
Pages 62-75 | Received 20 April 2021, Accepted 12 June 2021, Published 01 June 2022
Abstract
A full statement and a rather detailed proof are given of a folklore result describing the category of (unital) algebras over a finite direct product of commutative rings. Following an extensive survey of some recent work on minimal ring extensions and chain conditions for (unital) ring extensions such as the FCP and FMC properties, including generalizations of these conditions and the FIP property to ring extensions involving noncommutative rings, a corollary of the folklore theorem for FIP, FCP and FMC is given for ring extensions
A \subseteq B where A is a finite direct product of commutative rings and B is a (not necessarily commutative) A-algebra.
Keywords: Commutative ring, unital algebra, finite direct product, categorical equivalence, minimal ringextension, FMC, FCP, FIP.
MSC numbers: Primary 16B50; Secondary 13B99, 18A05.
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