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.
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.) |