Pseudo-almost valuation domains are quasilocal going-down domains,but not conversely |
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Authors: | David E Dobbs |
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Affiliation: | (1) Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300, USA |
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Abstract: | Abstract Ayman Badawi has recently introduced the PAVDs, a class of (commutative integral) domains which is found strictly between
the class of APVDs (“almost pseudo valuation domains”) and that of the (necessarily quasilocal) domains having a linearly
ordered prime spectrum. It is known that the latter class strictly contains the class of quasilocal going-down domains; it
is proved that the class of quasilocal going-down domains strictly contains the class of PAVDs. Consequently, each seminormal
PAVD is a divided domain. Moreover, for each n, 1 ≤ n ≤ ∞, an example is constructed of a divided domain (necessarily a quasilocal going-down domain) of Krull dimension n which is not a PAVD.
Keywords Pseudo-almost valuation domain, Prime ideal, Going-down domain, Divided domain, Quasilocal, Valuation overring, Root extension,
Seminormal, D+M construction, Krull dimension
Mathematics Subject Classification (2000) Primary 13B24, 13G05, Secondary 13A15, 13F05 |
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