zbMATH — the first resource for mathematics

Dimension de Krull des anneaux de séries formelles sur un produit fibré. (Krull dimension of formal power series rings on a fibre product). (French) Zbl 0748.13005
Let \(T\) be a (commutative but not necessarily Noetherian) local integral domain, with maximal ideal \({\mathcal M}\) and residue field \(K\). Let \(\varphi:T\to K\) be the canonical homomorphism and let \(D\) be a proper subring of \(K\). The author considers the Krull dimension of \(R[[X]]\), where \(R=\varphi^{-1}(D)\). J. T. Arnold [Trans. Am. Math. Soc. 177, 299-304 (1973; Zbl 0262.13007)] has introduced the concept of SFT- ring, and proved that if a commutative ring \(A\) is not an SFT-ring, then \(\dim A[[X]]\) is infinite. Some sample results from the present paper are the following:
(a) \(R\) is an SFT-ring if and only if \(T\) and \(D\) are both SFT-rings; (b) if \(T\) is Noetherian, or if \(T\) is a discrete valuation ring (with value group possibly of rank greater than one), or if \(D\) is a field, then \(\dim R[[X]]=\dim D[[X]]+\dim T[[X]]-1\).
Using this result the author produces an example of a domain \(R\) of Krull dimension \(n\) such that: (1) \(\dim R[[X]]=\dim R+1\); (2) \(R[[X]]\) is catenary; (3) \(R\) is neither Noetherian nor a discrete valuation ring. Thus \(R\) is a new type of ring with properties (1) and (2).

13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13F25 Formal power series rings
Full Text: DOI
[1] Anderson D.F., Bouvier A., Dobbs D.E., Fontana M., Kabbaj S.,On Jaffard Domains, Expositiones Mathematicae6 (1988), 145–175.
[2] Arnold J.,Krull Dimension in Power Series Rings, Trans. Amer. Math. Soc.177 (1973), 299–304. · Zbl 0262.13007 · doi:10.1090/S0002-9947-1973-0316451-8
[3] Arnold J.,Power Series Rings over Prüfer Domains, Pacific J. Math.44 (1973), 1–11. · Zbl 0223.13016
[4] Arnold J.,Power Series Rings with finite Krull Dimension, Ind. Univ. Math. J.31 n. 6, (1982), 897–911. · Zbl 0498.13008 · doi:10.1512/iumj.1982.31.31061
[5] Arnold J.,The catenarian Property of Power Series Rings over Prüfer Domains, Proc. Amer. Math. Soc.94 (1985). · Zbl 0577.13010
[6] Brewer J.W.,Power Series Rings over Commutative Rings, Marcel Dekker, INC., (1983). · Zbl 0512.13013
[7] Fields D.E.,Dimension Theory in Power Series Rings, Pacific J. Math.35 (1970), 601–611. · Zbl 0192.38701
[8] Girolami F.,Power Series Rings over Globalized Pseudo-Valuation Domaines, J. of Pure and App. Algebra50 (1988), 259–269. · Zbl 0664.13007 · doi:10.1016/0022-4049(88)90104-1
[9] Nachar G.,Sur la caténarité des anneaux de séries formelles sur un anneau de Prüfer, Communications in Agebra15 (3), (1987), 479–489. · Zbl 0632.13006 · doi:10.1080/00927878708823429
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.