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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