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The catenarian property of the polynomial rings over a Prüfer domain. (English) Zbl 0577.13011
Sémin. d’algèbre P. Dubreil et M.-P. Malliavin, 36ème Année, Proc., Paris 1983/84, Lect. Notes Math. 1146, 340-354 (1985).
[For the entire collection see Zbl 0562.00001.]
This paper is concerned with the catenarian property for polynomial rings over Prüfer domains. Its starting point is the following question posed by P. Ribenboim: if V is a valuation domain of finite dimension, is the polynomial ring \(V[T_ 1,...,T_ n]\) catenarian? A paper by A. M. de Souza Doering and Y. Lequain [J. Algebra 78, 163-180 (1982; Zbl 0496.13006)] contains the result that if R is a finite-dimensional Prüfer domain, then R[T] is catenarian. Among the main results of the paper under review are that if V is a 1-dimensional valuation domain then the polynomial ring \(V[T_ 1,...,T_ n]\) is catenarian for every positive integer n (corollary 10), and that if R is a Prüfer domain such that \(\dim (R_{{\mathfrak m}})\) is finite for every maximal ideal \({\mathfrak m}\) of R, then \(R[T_ 1,...,T_ n]\) is catenarian for every positive integer n (theorem 12). The main techniques used are pull-backs and a function introduced to measure the extent to which prime ideals in polynomial domains fail to be extended.
Reviewer: R.Y.Sharp

MSC:
13E99 Chain conditions, finiteness conditions in commutative ring theory
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13G05 Integral domains
13B02 Extension theory of commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)