×

zbMATH — the first resource for mathematics

On the elasticities of Krull domains with finite cyclic divisor class group. (English) Zbl 0964.13001
Author’s abstract: Let \(R\) be a Krull domain with finite divisor class group \(\text{Cl}(R)\). We consider possible values of \(\rho(R)\), the elasticity of factorizations of \(R\). We first determine an upper bound on \(\rho(R)\) based on the distribution of height-one prime ideals in \(\text{Cl} (R)\) and characterize when this upper bound is attained. We concentrate on the case \(\text{Cl}(R) =\mathbb{Z}_{p^k}\), where \(p\) is a prime, and determine further bounds on \(\rho(R)\) when \(k=1\) (i.e., \(\text{Cl} (R)=\mathbb{Z}_p)\). Unlike a related analysis for the cross number of \(\mathbb{Z}_{p^k}\), we show that the elasticities of such domains do not take on a complete set of hypothesized values.

MSC:
13A05 Divisibility and factorizations in commutative rings
13C20 Class groups
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1016/0022-4049(92)90144-5 · Zbl 0773.13003
[2] DOI: 10.1090/S0002-9939-1993-1106176-1
[3] Anderson D.F., Fac-torization in Integral Domains 189 pp 1– (1997)
[4] Chapman S.T., Zero-dimensional Commutative Rings 171 pp 167– (1995)
[5] Chapman S.T., Australasian J. Comb 14 pp 85– (1996)
[6] Chapman S.T., Factorization in Integral Domains 189 pp 73– (1997)
[7] DOI: 10.1080/00927879208824442 · Zbl 0761.13010
[8] DOI: 10.1006/jabr.1993.1153 · Zbl 0784.11050
[9] DOI: 10.1112/S0025579300015060 · Zbl 0757.13007
[10] Chapman S.T., Colloq. Math 70 pp 219– (1996)
[11] Geroldinger A., Col. Math. Soc. János Bolyai 51 pp 723– (1987)
[12] Geroldinger A., Abh. Math Sem 60 pp 115– (1990)
[13] DOI: 10.1017/S0004972700044063
[14] DOI: 10.1007/BF01161801 · Zbl 0522.12006
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.