Existence of Erdős-Burgess constant in commutative rings

2020/05/17 by Guoqing Wang, Wang, Guoqing
Mathematics · #05E40 #11B75 #20M13 #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.2005.08955

openalex publication_date 2020/05/17 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28

Abstract

Let R be a 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 (if exists) such that for any given ℓ elements (not necessarily distinct) 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 prove that except for an infinite commutative ring with a very special form, the Erdős-Burgess constant of the ring R exists if and only if R is finite.

Citations

• On a Conjecture of Littlewood and Idempotent Measures

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