W-Jaffard Domains in Pullbacks

Abstract

Abstract. In this paper we study the class of w-Jaffard domains in pullback constructions, and give new examples of these domains. In particular we give examples to show that the two classes of w-Jaffard and Jaffard domains are incomparable. As another application, we establish that for each pair of positive integers (n, m) with n + 1 ≤ m ≤ 2n + 1, there is an (integrally closed) integral domain R such that w-dim(R) = n and w[X]-dim(R[X]) = m. IntroductionThroughout this paper, R denotes a (commutative integral) domain with identity with quotient field qf (R), and let X be an algebraically independent indeterminate over R. In [26, Theorem 2] Seidenberg proved that if R has finite Krull dimension, Krull [18] has shown that if R is any finite-dimensional Noetherian ring, then dim(R[X]) = 1 + dim(R) (cf. also [26, Theorem 9]). Seidenberg subsequently proved the same equality in case R is any finite-dimensional Prüfer domain. To unify and extend such results on Krull-dimension, Jaffard [17] introduced and studied the valuative dimension denoted by dim v (R), for a domain R. This is the maximum of the ranks of the valuation overrings of R. Jaffard proved in [17, Chapitre IV] that, if R has finite valuative dimension, then dim v (R[X]) = 1 + dim v (R), and that if R is a Noetherian or a Prüfer domain, then dim(R) = dim v (R). In [1] Anderson, Bouvier, Dobbs, Fontana and Kabbaj introduced the notion of Jaffard domains, as finite dimensional integral domains R such that dim(R) = dim v (R), and studied this class of domain systematically (see also [6]).The v, t and w-operations in integral domains are of special importance in multiplicative ideal theory and was investigated by many authors in the 1980's. Ideal w-multiplication converts ring notions such as Dedekind, Noetherian, Prüfer, and quasi-Prüfer, respectively to Krull, strong Mori, PvMD, and UMt. As the wcounterpart of Jaffard domains, in [22], we introduced the class of w-Jaffard domains, as integral domains R such that w-dim(R) = w-dim v (R) < ∞. In this paper we study the transfer of w-Jaffard domains in pullback constructions, in order to provide original examples.We need to recall some notions from star operations. Let F (R) denotes the set of nonzero fractional ideals, and f (R) be the set of all nonzero finitely generated fractional ideals of R. Let * be a star operation on the domain R. For every A ∈ F (R), put A * f := F * , where the union is taken over all F ∈ f (R) with2000 Mathematics Subject Classification. Primary 13G05, 13A15, 13C15.

On Quasi-Prüfer and UMtDomains

“…[1]). It is proved in [16], that D is a * -Jaffard domain if and only if * [X 1 , · · · , X n ]-dim(D[X 1 , · · · , X n ]) = * -dim(D) + n, for each positive integer n. In [19] we gave examples to show that the two classes of w-Jaffard and Jaffard domains are incomparable by constructing a w-Jaffard domain which is not Jaffard and a Jaffard domain which is not w-Jaffard.…”