A treed domain need not be valtreed
David E. Dobbs
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320
Pages 265-278 | Received 09 January 2024, Accepted 17 February 2024, Published 08 November 2024
Abstract
If 3 \leq n \leq \infty, there exists a quasi-local treed domain which has Krull dimension n and is not a valtreed domain. A consequence is that the class of valtreed domains fits properly between the class of treed domains and the class of going-down domains. Although the class of treed domains that are not going-down domains is stable under the classical D+M construction, the class of valtreed domains that are not going-down domains is markedly unstable under that construction.
Keywords: Commutative ring, integral domain, prime ideal, treed domain, valuation domain, maximal ideal, valtreed domain, going-down domain, field extension, integral closure, left topology, D+M construction.
MSC numbers: Primary 13B21, 13G05; Secondary 13A15, 13F05, 54A10.
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