Sunto
Si introducono i domini di pseudo-valutazione di tipon, per ognin>-0, i quali risultano particolari domini divisi, nel senso di Akiba [1] e Dobbs[8], e generalizzano i «classici» domini di pseudo-valutazione di Hedstrom-Houston [19]. Si dimostrano varie proprietà di questi anelli, quali quella di essere seminormali, ma non normali, in generale, e quella di essere »generati» iterando un’operazione di prodotto fibrato di anelli di pseudo-valutazione (che estende l’operazione di composizione di anelli di valutazione, cfr. Nagata [25, p.35]) I principali risultati qui ottenuti consistono in una caratterizzazione degli anelli di pseudo-valutazione di tipon tali che l’anello dei polinomi è catenario e nella possibilità di costruireesplicitamente un anello (di pseudo-valutazione di tipon opportuno),R, tale che, presi comunque due interid≥e≥0, risulti dim (R)=d, dim (R[X])=d+e+1 (cfr. Seidenberg [29]).
Summary
We introduce the pseudo-valuation domains of typen (briefly,P n VD), for everyn≥0. They are special divided domains (cf. Akiba [1], Dobbs [8]) and generalize the «classical» pseudo-valuation domains (cf. Hedstrom-Houston [19]). We prove several properties concerning these domains: e.g. everyP n VD is seminormal, but in general not normal; everyP n VD can be constructed by a sequence of pull-backs, preserving the flavour of the composition of valuation rings (cf. Nagata [25, p. 35]). The main results of the present paper are the following: (a) a characterization of theP n VDs A such that the polynomial ringA[X] is catenarian; (b) anexplicit construction of a class of integral domains (in fact,P n VDs)R such that, for everyd≥e≥0, dim (R=d and dim (R[X])=d+e+1 (cf. Seidenberg [29]).
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(Conferenza tenuta il 3 novembre 1981)
Classificazione AMS (MOS) 1980: 13 A 17, 13 A 18, 13 B 25, 13 C 15, 13 F 05, 13 F 25.
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Fontana, M. Sur quelques classes d’anneaux divisés. Seminario Mat. e. Fis. di Milano 51, 179–200 (1981). https://doi.org/10.1007/BF02924821
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DOI: https://doi.org/10.1007/BF02924821