Abstract
This paper gives complete proofs of the following result: let R be a locally finite dimensional Prüfer domain; then, the polynomial ring R[T1,..,Tr] is catenarian for every r⩾1. The main techniques used in the proof are pull-backs and a function introduced here to measure the extent to which prime ideals in polynomial domains fail to be extended.
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Bibliographie
N. BOURBAKI, Algèbre commutative. Chap 7, Hermann Paris (1961).
A. BOUVIER, F. BURQ et G. GERMAIN. Un exemple d'anneau caténaire. Pub. Dépt. Math. Lyon, 17–1, 1980, 97–101.
A. BOUVIER, M. CONTESSA and P. RIBENBOIM. On chains of prime ideals in polynomial rings. C.R. Math. Rep. Acad. Sci. Canada 3, (1981), 87–92.
A. BOUVIER, M. FONTANA. On the catenarian property of the polynomial rings over a Prüfer domain. C.R. Math. Rep. Acad. Sci. Canada, 4, (1983), p. 97–100.
J.W. BREWER, P.R. MONTGOMERY, E.A. RUTTER and W.J. HEINZER. Krull dimensions of polynomial rings. Lecture Notes in Math. no311 (1973), 26–43, Springer-Verlag.
A.M. DE SOUZA DOERING and Y. LEOUAIN. Chains of prime ideals in polynomial rings. J. Algebra 78, (1982), 163–180.
D.E. DOBBS. Divided rings and going-down. Pacif. J. Math. 67, (1976), 353–363.
D.E. DOBBS and M. FONTANA. On pseudo-valuation domains and their globalisations. Proc. Trento Conference, Dekker Lect. Notes N. 84, 65–77, 1983.
M. FONTANA. Topologically defined classes of commutative rings. Ann. Mat. Pura Appl. 123, (1980), 331–335.
M. FONTANA. Sur quelques classes d'anneaux divisés. Rend. Sem. Mat. Fis. Milano 51 (1981), p.173–200.
T. GERAMITA and C. SMALL. Introduction to homological methods in commutative rings. Queen's papers in pure and applied math. no 43, (1976).
P. JAFFARD. Théorie de la dimension dans les anneaux de polynômes. Gauthier-Villars, Paris (1960).
I. KAPLANSKY. Commutative rings. Univ. Chicago Press, 2nd. edit. (1974).
Y. LEQUAIN. Catenarian property in a domain of formal power series. J. Algebra 65, (1980), 110–117.
S. MALIK and J.L. MOTT. Strong S-domains, Journal of pure and applied algebra 28 no3 (1983), p. 249–265.
H. MATSUMURA. Commutative algebra. Benjamin 2nd Edit. (1980).
M. NAGATA. Finitely generated rings over a valuation ring, J. Math. Kyoto Univ. 5–2 (1966) p. 163–169.
S. McADAM. Going down in polynomial rings. Can. J. Math. 23, (1971), 704–711.
L. RATLIFF. On quasi-unmixed local domains, the altitude formula and the chain condition for prime ideals (I). Amer. J. Math. 51, (1969), 508–528.
A. SEINDENBERG. A note on the dimension theory of rings. Pacif. J. Math., 3, (1953), 505–512.
H. SEYDI. Anneaux henséliens et conditions de chaînes. Bull. Soc. Math. France, 38, (1970), 9–31.
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Bouvier, A., Fontana, M. (1985). The catenarian property of the polynomial rings over a Prüfer domain. In: Malliavin, MP. (eds) Séminaire d'Algèbre Paul Dubreil et Marie-Paule Malliavin. Lecture Notes in Mathematics, vol 1146. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074546
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DOI: https://doi.org/10.1007/BFb0074546
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