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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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