Numbers of generators of ideals in a group ring of an elementary Abelian \(p\)-group
type: Item
instance of: scholarly article
title: Numbers of generators of ideals in a group ring of an elementary Abelian \(p\)-group
zbMATH Open document ID: 0946.13014
author: J. S. Okon
author: David E. Rush
author: J. Paul Vicknair
published in: Journal of Algebra
publication date: 2000-06-29T00:00:00Z
review text: Let \(\mu(I)\) denote the minimal number of generators of an ideal \(I\) of a local ring \((R,M)\). The numbers \(d(R) = \max \{ \mu(I) :I\text{ is an ideal of } R \}\) and \( \operatorname {sp}(R) = \max \{ \mu (M^{i}): i \geq 0 \}\) are called, respectively, the Dilworth and the Sperner number of \(R\). The paper is mainly devoted to prove that if \((A,m)\) is an Artin local ring of characteristic \(p^{n}\), \(n > 1\), \(m = zA\) with \(pA = z^{e}A\), \(G=(\mathbb{Z}/p\mathbb{Z})^{k}\), \(\alpha = \min \{e, \varphi(p)\}\) and \(m^{(k-1) \varphi(p) + \alpha} \neq 0\), then the Dilworth and Sperner numbers of the group ring \(A[G]\) coincide and they are equal to \(1 + \frac{\alpha}{\varphi(p)} (|G|-1)\), where \(\varphi\) denotes the Euler function. Moreover, the authors show that whenever \((R,m)\) is a \(1\)-dimensional local Noetherian ring with \(R/m\) of characteristic \(p\) and \(G\) is a finite Abelian \(p\)-group, then the Dilworth numbers of \(R[G]\) and of \((R/m^{c})[G]\) coincide for \(c\) large, and so it happens for the Sperner numbers. The authors apply this to obtain the Dilworth and Sperner numbers of the group ring \(A[G]\) for certain one-dimensional group rings.
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 1423892
zbMATH Keywords: Dilworth number
zbMATH Keywords: Sperner number
zbMATH Keywords: number of generators of an ideal
zbMATH Keywords: Artin local ring
reviewed by: Carlos Galindo
MaRDI profile type: Publication
full work available at URL: jabr.1999.8082
OpenAlex ID: W2011846214
cites work: Linear maps over abelian group algebras
cites work: Group rings R[G] with 3-generated ideals when R is artinian
cites work: Group rings \(R[G]\) with \(4\)-generated ideals when \(R\) is an Artinian principal ideal ring
cites work: The dilworth number of group rings over an artin local ring
cites work: On the ubiquity of Gorenstein rings
cites work: The dimensions of periodic modules over modular group algebras
cites work: Q4721262
cites work: Q5805073
cites work: Units in regular elementary Abelian group rings
cites work: Units and class-groups in integral elementary Abelian group rings
cites work: On the structure of principal ideal rings
cites work: Q4885654
cites work: Preservation of local properties and chain conditions in commutative group rings
cites work: Local rings with bounded ideals
cites work: Commutative Semigroup Rings with Two-Generated Ideals
cites work: Two-generated ideals in non-Noetherian semigroup rings
cites work: Group rings with n-generated ideals
cites work: A Gorenstein Ring with Larger Dilworth Number than Sperner Number
cites work: Numbers of generators of ideals in zero-dimensional group rings
cites work: Rings with two-generated ideals
cites work: Two-generated ideals and representations of Abelian groups over valuation rings
cites work: Q4177688
cites work: Q3792779
cites work: The Dilworth number of Artin Gorenstein rings
DOI: 10.1006/JABR.1999.8082
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description: scientific article; zbMATH DE number 1423892