A going-down theoretic proof that straight domains are locally divided
David E. Dobbs
DepartmentofMathematics,UniversityofTennessee,Knoxville,Tennessee37996-1320.
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Pages 15-25 | Received 10 March 2025, Accepted 06 June 2025, Published 19 July 2026
Abstract
A result on going-down domains is used to give a new proof of the recent result of Spencer Secord that all straight domains are locally divided domains. Corollaries collect characterizations of straight domains (equivalently, of locally divided domains) and of prime ideals of a base domain R that are straight with respect to all overring extensions of R. For integrally closed (but not necessarily quasi-local) base domains R, it is shown how to use the going-down to P” property and the behavior of a relatively small class of singly generated overrings of R in order to characterize the locally divided prime ideals of R, while avoiding any mention ofstraight”-related concepts.
Keywords: Integral domain, prime ideal, going-down, going-down domain, divided domain, locally divided domain, torsion-free module, quotient field, overring, straight domain.
MSC numbers: Primary 13B21, 13F05; Secondary 13G05, 13A15, 13C13.
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