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AIMS Mathematics

2020, Volume 5, Issue 5: 4424-4431. doi: 10.3934/math.2020282
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Research article

Lower bound for the Erdős-Burgess constant of finite commutative rings

  • Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, P. R. China
  • Received: 20 March 2020 Accepted: 07 May 2020 Published: 12 May 2020

  • MSC : 05E40, 11B75, 13M99, 20M25

  • Let $R$ be a finite commutative unitary ring. An idempotent in $R$ is an element $e\in R$ with $e^2 = e$. The Erdős-Burgess constant associated with the ring $R$ is the smallest positive integer $\ell$ such that for any given $\ell$ elements (repetitions are allowed) of $R$, say $a_1, \ldots, a_{\ell}\in R$, there must exist a nonempty subset $J\subset \{1, 2, \ldots, \ell\}$ with $\prod\limits_{j\in J} a_j$ being an idempotent. In this paper, we give a lower bound of the Erdős-Burgess constant in a finite commutative unitary ring in terms of all its maximal ideals, and prove that the lower bound is attained in some cases. The result unifies some recently obtained theorems on this invariant.

    Citation: Guoqing Wang. Lower bound for the Erdős-Burgess constant of finite commutative rings[J]. AIMS Mathematics, 2020, 5(5): 4424-4431. doi: 10.3934/math.2020282

    Related Papers:

  • Let $R$ be a finite commutative unitary ring. An idempotent in $R$ is an element $e\in R$ with $e^2 = e$. The Erdős-Burgess constant associated with the ring $R$ is the smallest positive integer $\ell$ such that for any given $\ell$ elements (repetitions are allowed) of $R$, say $a_1, \ldots, a_{\ell}\in R$, there must exist a nonempty subset $J\subset \{1, 2, \ldots, \ell\}$ with $\prod\limits_{j\in J} a_j$ being an idempotent. In this paper, we give a lower bound of the Erdős-Burgess constant in a finite commutative unitary ring in terms of all its maximal ideals, and prove that the lower bound is attained in some cases. The result unifies some recently obtained theorems on this invariant.


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    [1] D. A. Burgess, A problem on semi-groups, Studia Sci. Math. Hungar., 4 (1969), 9-11.
    [2] C. Deng, Davenport constant for commutative rings, J. Number Theory, 172 (2017), 321-342.
    [3] W. Gao, A. Geroldinger, Zero-sum problems in finite abelian groups: A survey, Expo. Math., 24 (2006), 337-369.
    [4] W. Gao, Y. Li, J. Peng, An upper bound for the Davenport constant of finite groups, J. Pure Appl. Algebra, 218 (2014), 1838-1844.
    [5] A. Geroldinger, D. J. Grynkiewicz, The large Davenport constant Ⅰ: Groups with a cyclic, index 2 subgroup, J. Pure Appl. Algebra, 217 (2013), 863-885.
    [6] A. Geroldinger, M. Liebmann, A. Philipp, On the Davenport constant and on the structure of extremal zero-sum free sequences, Period. Math. Hungar., 64 (2012), 213-225.
    [7] D. W. H. Gillam, T. E. Hall, N. H. Williams, On finite semigroups and idempotents, Bull. London Math. Soc., 4 (1972), 143-144.
    [8] P. A. Grillet, Commutative Semigroups, Kluwer Academic Publishers, 2001.
    [9] J. Hao, H. Wang, L. Zhang, On the modular Erdős-Burgess constant, Open J. Discrete Math., 9 (2019), 11-16.
    [10] N. Kravitz, A. Sah, A stronger connection between the Erdős-Burgess and Davenport constants, J. Number Theory, 210 (2020), 373-388.
    [11] J. E. Olson, A combinatorial problem on finite Abelian groups, I, J. Number Theory, 1 (1969), 8-10.
    [12] G. Wang, Davenport constant for semigroups II, J. Number Theory, 153 (2015), 124-134.
    [13] G. Wang, Additively irreducible sequences in commutative semigroups, J. Combin. Theory Ser. A, 152 (2017), 380-397.
    [14] G. Wang, Structure of the largest idempotent-product free sequences in semigroups, J. Number Theory, 195 (2019), 84-95.
    [15] G. Wang, Erdős-Burgess constant of commutative semigroups, arXiv: 1802.08791.
    [16] H. Wang, J. Hao, L. Zhang, On the Erdős-Burgess constant of the multiplicative semigroup of a factor ring of $\mathbb{F}_q[x]$, Int. J. Number Theory, 15 (2019), 131-136.
    [17] H. Wang, L. Zhang, Q. Wang, et al. Davenport constant of the multiplicative semigroup of the quotient ring $\frac{\mathbb{F}_p[x]}{\langle f(x)\rangle}$, Int. J. Number Theory, 12 (2016), 663-669.
    [18] L. Zhang, H. Wang, Y. Qu, A problem of Wang on Davenport constant for the multiplicative semigroup of the quotient ring of $\mathbb{F}_2[x]$, Colloq. Math., 148 (2017), 123-130.
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