TREED DOMAINS

Year 2008, Volume: 3 Issue: 3, 43 - 57, 01.06.2008

Gabriel Picavet

Abstract

We establish that treed domains are well behaved in Zafrullah’s sense and have locally polynomial depth 1. For the DW-domains R of Mimouni, such that I−1 6= R for each nontrivial finitely generated ideal I of R, likewise results are proven. We study some special treed domains and show in particular that the Nagata ring of an integral domain R is (locally) divided if and only if R is (locally) divided and quasi-Prüfer. We show that the small finitistic dimension of a local treed domain is 1 and calculate the small finitistic dimension of localizations of polynomial rings over a treed domain.

Keywords

DW-domain, divided domain, going-down domain, H-domain, idomain, Nagata ring, (polynomial) grade, quasi-Prüfer domain, small finitistic dimension, t-ideal, treed domain, well behaved prime

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• Universit´e Blaise Pascal,
• Aubi`ere Cedex, France
• E-mail: [e-mail]