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Davenport constant for commutative rings
Open accessAbstract
The Davenport constant is one measure for how "large" a finite abelian group is. In particular, the Davenport constant of an abelian group is the smallest $k$ such that any sequence of length $k$ is reducible. This definition extends naturally to commutative semigroups, and has been studied in certain finite commutative rings. In this paper, we give an exact formula for the Davenport constant of a general commutative ring in terms of its unit group.
Overview
CoChat Summary
TL;DR
The paper gives an exact formula for the Davenport constant \(D(R)\) of a finite commutative ring \(R\) (as a multiplicative semigroup) in terms of its unit group and certain index-2 phenomena, after reducing to local rings. It also proves \(D(R)<\infty\) iff \(R\) is finite, bounds \(D(R)-D(U(R))\) by the number of index-two ideals, and classifies local rings with nonzero difference.
Key findings
- Theorem 3.2 gives an exact formula for \(D(R)\) when \(R\) decomposes as a product of copies of \( \mathbb Z/2\mathbb Z\), \( \mathbb Z/4\mathbb Z\), \( \mathbb Z/8\mathbb Z\), \( \mathbb F_2[x]/(x^2)\), and a product \(R'\) of local rings avoiding those four; the formula is expressed via \(D(U(R))\) and Davenport constants of certain subgroups, with well-defined quantities \(k_1,k_2,k_3,k_4\).
- Corollary 3.2: for finite commutative \(R\), \(D(R) \le D(U(R)) + n_2(R)\), where \(n_2(R)\) is the number of index-two (prime, maximal) ideals of \(R\). Corollary 3.1 gives a sharper bound on \(D(R)-D(U(R))\), with equality on the right if \(U(R)\) is a power of 2 or if \(k_i=0\) for \(i=2,3,4\).
- Theorem 9.2: for a finite local ring \(R\), \(\Delta(R)=D(R)-D(U(R)) \le 1\), with equality iff \(R\) is one of \( \mathbb Z/2\mathbb Z\), \( \mathbb Z/4\mathbb Z\), \( \mathbb Z/8\mathbb Z\), or \( \mathbb F_2[x]/(x^2)\). The paper’s table gives \(D(R)/D(U(R))\) as \(2/1\), \(3/2\), \(4/3\), and \(3/2\), respectively.
- Theorem 3.1: if \(R\) is commutative and \(D(R)<\infty\), then \(|R|<\infty\). Theorem 11.1 lists cases where \(D(R)=D(U(R))\), including \(|R|\) odd, \(R\) an \(\mathbb F_q\)-algebra with \(q\) a power of 2 other than 2, and \(R=\mathbb Z/n\mathbb Z\) with \(16\mid n\). Theorem 11.2 refines the Zhang–Wang–Qu bound for \( \mathbb F_2[x]/(f)\), and Theorem 11.3 treats \(R=\mathbb Z/n_1\mathbb Z \times \cdots \times \mathbb Z/n_r\mathbb Z\).
Methods
- Theoretical additive number theory and commutative algebra; no empirical sample or data. The paper introduces the relative Davenport constant \(D(S,T)\), unit-stabilized pairs, and almost unit-stabilized semigroups, then proves gluing theorems for products of semigroups.
- The main proof reduces a general finite commutative ring to finite local rings via its Artinian product decomposition, classifies local rings as unit-stabilized when the residue field is not \(\mathbb F_2\) and almost unit-stabilized when it is \(\mathbb F_2\), and uses lemmas involving Nakayama’s lemma, Chinese remainder, and stabilization of annihilators.
- Main analysis consists of exact formulas and inequalities for \(D(R)\), \(D(U(R))\), and \(\Delta(R)\), followed by corollaries and special-case computations.
Limitations
- The paper states that it still does not have a complete classification of finite rings \(R\) with \(D(R)=D(U(R))\), even for \(R=\mathbb Z/n\mathbb Z\) or \(R=\mathbb F_2[x]/(f)\); the general cases \(n=4b,8b\) remain open, as does the non-squarefree \(f\) case.
- The paper states that these questions depend on the relation between \(D(G)\) and \(D(G \times C_2^e)\) for abelian groups \(G\), and that very little is known about \(D(G)-M(G)\) for general \(G\) of large rank.
- Evident from the provided text: several exact formulas and bounds are not fully rendered, so their precise algebraic expressions cannot be quoted here.
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0 Valid · 4 Not valid · 14 Not checked · 68 Own results and interpretations
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“which means that m is principal.”
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In the paper
“Theorem 11.1. D ( R ) = D ( U ( R )) in any of the following scenarios:”
- Also stated in: § 11. A few more corollaries of Theorem 3.2 · p. 13 · plain
“- (5) R is of the form Z / 4 Z × R ′ or F 2 [ x ] / ( x 2 ) × R ′ , where | U ( R ′ ) | > 1 is odd and ∆( R ′ ) = 0 .”
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- Also stated in: § 11. A few more corollaries of Theorem 3.2 · p. 13 · plain
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“It turns out local rings are all either unit-stabilized or almost unit-stabilized.”
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“Theorem 11.1. D ( R ) = D ( U ( R )) in any of the following scenarios:”
- Also stated in: § 11. A few more corollaries of Theorem 3.2 · p. 13 · plain
“- (2) R is an F q algebra, where q is a power of 2 other than 2.”
plain- Asserted: passes — The authors put it forward as true.
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- Also stated in: § 11. A few more corollaries of Theorem 3.2 · p. 13 · plain
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“Thus in the additive group of R , 1 has order 2 n , which means that we must have R ≃ Z / 2 n Z .”
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In the paper
§ 10. Proof of Theorem 3.2 · p. 13
“- U ( R 1 ) = 0, D ( R 1 ) = 1. - U ( R 3 ) ≃ C 2 × C 2 , D ( R 3 ) = 3. - U ( R 2 ) ≃ C 2 , D ( R 2 ) = 2. - U ( R 4 ) ≃ C 2 , D ( R 4 ) = 2.”
§ Theorem 9.1. We have: · p. 12
“| Z / 2 Z | 2 | 1 | | Z / 4 Z | 3 | 2 | | Z / 8 Z | 4 | 3 | | F 2 [ x ] / ( x 2 ) | 3 | 2 |”
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In the paper
§ Lemma 5.1. Suppose G is an infinite abelian group. Then D ( G ) = ∞ . · p. 5
“Then g 1 , . . . , g n , g is not irreducible if and only if g -1 is not of the form g i 1 · · · g i k .”
§ Lemma 5.1. Suppose G is an infinite abelian group. Then D ( G ) = ∞ . · p. 5
“However, there are at most 2 n such g , which means that we can always extend an irreducible sequence to get a longer irreducible sequence”
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In the paper
§ 8. Almost Unit-Stabilized semigroups · p. 10
“- S 1 , S, S ′ are unit-stabilized.”
§ 7. Unit-Stabilized Pairs · p. 7
“However, if R has residue field F 2 , then it is in fact not unit-stabilized: U ( R ) = 1 + m , where m is the maximal ideal of R .”
Citations
Cites
Introduction To Commutative Algebra
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Introduction to Commutative Algebra.
Lewis, D., McDonald, B. R., Atiyah, M. F., & MacDonald, I. G. (1970) · American Mathematical Monthly · doi:10.2307/2316241 · Cited by 1,709
Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory
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A combinatorial problem on finite Abelian groups, I
Olson, J. E. (1969) · Journal of Number Theory · doi:10.1016/0022-314x(69)90021-3 · Cited by 290
A combinatorial problem on finite Abelian groups, II
Olson, J. E. (1969) · Journal of Number Theory · doi:10.1016/0022-314x(69)90037-7 · Cited by 134
Non-unique factorizations : algebraic, combinatorial and analytic theory
Geroldinger, A., & Halter‐Koch, F. (2006) · Thammasat University Digital Collections · Cited by 75
Geroldinger, A., & Schneider, R. (1992) · Journal of Combinatorial Theory, Series A · doi:10.1016/0097-3165(92)90061-x · Cited by 59
A combinatorial problem in Abelian groups
Rogers, K. (1963) · Mathematical Proceedings of the Cambridge Philosophical Society · doi:10.1017/s0305004100037221 · Cited by 47
A combinatorial problem on finite abelian groups, 3
Boas, P. V. E., & Kruyswijk, D. (1969) · Data Archiving and Networked Services (DANS) · Cited by 39
A note on the growth of Davenport's constant
Mazur, M. (1992) · manuscripta mathematica · doi:10.1007/bf02567669 · Cited by 24
Research works printed but not linked: 3
- Baayen, 1969, (C2⊕C2⊕C2⊕C2n)!, Math. Centrum Amsterdam Afd. Zuivere Wisk., 1969
- G. Wang, W. Gao, Davenport constant of the multiplicative semigroup of the ring Zn1⊕⋯⊕Znr, arXiv e-prints, March 2016.
- L. Zhang, H. Wang, Y. Qu, A problem of Wang on Davenport constant for the multiplicative semigroup of the quotient ring of F2[x], arXiv e-prints, July 2015.
Cited by
Erdős-Burgess constant of commutative semigroups
Wang, G. (2022) · Communications in Algebra · doi:10.1080/00927872.2022.2045492 · Cited by 6
Structure of long idempotent-sum-free sequences over finite cyclic semigroups
Wang, G. (2020) · International Journal of Number Theory · doi:10.1142/s1793042121500123 · Cited by 1
A generalization of Kruyswijk-Olson theorem on Davenport constant in commutative semigroups
Wang, G. (2020) · AIMS Mathematics · doi:10.3934/math.2020193
Lower bound for the Erdős-Burgess constant of finite commutative rings
Wang, G. (2020) · AIMS Mathematics · doi:10.3934/math.2020282
Correction to: Davenport constant for semigroups
Wang, G. (2020) · Semigroup Forum · doi:10.1007/s00233-020-10133-2
Metadata
- Authors
- Deng, C.
- Corresponding author
- Deng, Calvin
- Venue
- Journal of Number Theory
- Publisher
- Elsevier BV
- Published
- 2017-03
- Volume
- 172
- Pages
- 321-342
- ISSN
- 0022-314X (ISSN-L)
- OpenAlex
- W2963364585
- Semantic Scholar
- 010239bd071d4f58d669b794885cad65725bc9e8
- Cited by
- 6
- References
- 13
- Citations by year
- 2022: 1 · 2021: 1 · 2020: 4
- FWCI
- 0.00
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- www.elsevier.com/tdm/userlicense/1.0/ (tdm)
- www.elsevier.com/legal/tdmrep-license (tdm)
- www.elsevier.com/open-access/userlicense/1.0/ (vor, 1461 d)
- doi.org/10.15223/policy-017 (stm-asf)
- doi.org/10.15223/policy-037 (stm-asf)
- doi.org/10.15223/policy-012 (stm-asf)
- doi.org/10.15223/policy-029 (stm-asf)
- doi.org/10.15223/policy-004 (stm-asf)
- Locations
- Journal of Number Theory (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (1358659)
- University of Minnesota Duluth
- National Sanitarium Association (H98230-13-1-0273)
- Topics
- Physical Sciences › Mathematics › Algebra and Number Theory › Rings, Modules, and Algebras
- Physical Sciences › Mathematics › Algebra and Number Theory › Analytic Number Theory Research
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Advanced Combinatorial Mathematics
- Keywords
- Davenport constant, commutative rings, finite abelian groups, commutative semigroups, unit groups
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