×

zbMATH — the first resource for mathematics

On the Krull dimension of \(\text{Int}(D)\) when \(D\) is a pullback. (English) Zbl 0899.13024
Cahen, Paul-Jean (ed.) et al., Commutative ring theory. Proceedings of the 2nd international conference, Fès, Morocco, June 5–10, 1995. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 185, 457-470 (1997).
Let \(D\) be a finite dimensional domain with quotient field \(K\). One is interested in the Krull dimension of the ring \(\text{Int} (D):= \{f\in K[X]: f(D) \subset D\}\) of integer valued polynomials. If \(D\) has a nonzero ideal in common with an overring \(B\) relations between \(\dim \text{Int} (D)\) and \(\dim B\) are established. In particular an explicit formula for \(\dim \text{Int} (D)\) is obtained for a locally finite domain \(D\) (i.e. every nonzero element of \(D\) is contained only in finitely many maximal ideals) which is locally a pullback of a Jaffard domain [cf. D. F. Anderson, A. Bouvier, D. E. Dobbs, M. Fontana and S. Kabbaj, Expo. Math. 6, No. 2, 145-175 (1988; Zbl 0657.13011)]. Seidenberg’s construction [A. Seidenberg, Pac. J. Math. 4, 603-614 (1954; Zbl 0057.26802)] of domains \(D\) with different dimensions of \(D[X]\) is transferred to the rings \(\text{Int} (D)\). In case of pseudo-valuation domains of type \(n\) [cf. D. E. Dobbs and M. Fontana, C. R. Acad. Sci., Paris, Sér. I 306, No. 1, 11-16 (1988; Zbl 0642.13010)] a precise formula for \(\dim \text{Int} (D)\) is given.
For the entire collection see [Zbl 0855.00015].

MSC:
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13G05 Integral domains
13B25 Polynomials over commutative rings
PDF BibTeX XML Cite