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

2020/02/26 by Guoqing Wang, Wang, Guoqing
Mathematics · #05E40 #11B75 #13M99 #20M25 #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2002.11489

openalex publication_date 2020/02/26 · openalex created_date 2020/03/06 · openalex updated_date 2026/07/28

Abstract

Let R be a finite commutative unitary ring. An idempotent in R is an element e∈ R with e²=e. The Erdős-Burgess constant associated with the ring R is the smallest positive integer ℓ such that for any given ℓ elements (repetitions are allowed) of R, say a₁,…,a_ℓ∈ R, there must exist a nonempty subset J⊂ \1,2,…,ℓ\ with ∏_{j∈ 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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