首页/论文/2005.08955

arXiv 2005.089552020-05-17math.COmath.NT

Existence of Erdős-Burgess constant in commutative rings

Guoqing Wang

备注：: 8 pages. arXiv admin note: text overlap with arXiv:2002.11489

英文摘要：

Let $R$ be a 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 $\ell$ (if exists) such that for any given $\ell$ elements (not necessarily distinct) 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 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

On the modular Erdős-Burgess constant

Let $n$ be a positive integer. For any integer $a$, we say that $a$ is idempotent modulo $n$ if $a^2\equiv a\pmod n$. The $n$-modular Erdős-Burgess constant is the smallest positive integer $\ell$ such that any $\ell$ integers contain one or more integers whose product is idempotent modulo $n$. We gave a sharp lower bound of the $n$-modular Erdős-Burgess constant, in particular, we determined the $n$-modular Erdős-Burgess constant in the case when $n$ is a prime power or a product of pairwise distinct primes.

2018-07-19math.COmath.NT

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

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 …

2020-05-09math.COmath.NT

Weighted Erdős-Burgess and Davenport constant in commutative rings

Let $R$ be a finite commutative unitary ring. An idempotent in $R$ is an element $e\in R$ with $e^2=e$. Let $Ψ$ be a subgroup of the group ${\rm Aut}(R)$ of all automorphisms of $R$. The $Ψ-$weighted Erdős-Burgess constant ${\rm I}_Ψ(R)$ is defined as the smallest positive integer $\ell$ such that every sequence over $R$ of length at least $\ell$ must contain a nonempty subsequence $a_1,\ldots, a_{r}$ such that $\prod\limits_{i=1}^r ψ_i(a_i)$ is one idempotent of $R$ where $ψ_1,\ldots,ψ_r\in Ψ$. In this paper, for the finite quotient ring of a Dedekind domain $R$, a connection is established b…

2022-02-24math.COmath.NT

Erdős-Burgess constant of commutative semigroups

Let $\mathcal{S}$ be a nonempty commutative semigroup written additively. An element $e$ of $\mathcal{S}$ is said to be idempotent if $e+e=e$. The Erdős-Burgess constant of the semigroup $\mathcal{S}$ is defined as the smallest positive integer $\ell$ such that any $\mathcal{S}$-valued sequence $T$ of length $\ell$ contain a nonempty subsequence the sum of whose terms is an idempotent of $\mathcal{S}$. We make a study of ${\rm I}(\mathcal{S})$ when $\mathcal{S}$ is a direct product of arbitrarily many of cyclic semigroups. We give the necessary and sufficient conditions such that ${\rm I}(\mat…

2020-05-17math.COmath.NT

A stronger connection between the Erdős-Burgess and Davenport constants

The Erdős-Burgess constant of a semigroup $S$ is the smallest positive integer $k$ such that any sequence over $S$ of length $k$ contains a nonempty subsequence whose elements multiply to an idempotent element of $S$. In the case where $S$ is the multiplicative semigroup of $\mathbb{Z}/n\mathbb{Z}$, we confirm a conjecture connecting the Erdős-Burgess constant of $S$ and the Davenport constant of $(\mathbb{Z}/n\mathbb{Z})^{\times}$ for $n$ with at most two prime factors. We also discuss the extension of our techniques to other rings.

2018-08-18math.CO

On the Erdős-Burgess constant of the multiplicative semigroup of a factor ring of $\mathbb{F}_q[x]$

Let $\mathcal{S}$ be a commutative semigroup endowed with a binary associative operation $+$. An element $e$ of $\mathcal{S}$ is said to be idempotent if $e+e=e$. The {\sl Erdős-Burgess constant} of $\mathcal{S}$ is defined as the smallest $\ell\in \mathbb{N}\cup \{\infty\}$ such that any sequence $T$ of terms from $S$ and of length $\ell$ contains a nonempty subsequence the sum of whose terms is idempotent. Let $q$ be a prime power, and let $\F_q[x]$ be the polynomial ring over the finite field $\F_q$. Let $R=\F_q[x]\diagup K$ be a quotient ring of $\F_q[x]$ modulo any ideal $K$. We gave a sh…

2018-07-18math.CO

更多相关论文收起更多论文

有限交换半群的Erdens-Ginzburg-Ziv定理

Erdős-Ginzburg-Ziv theorem for finite commutative semigroups

设$\mathcal{S}$是一个可加的有限交换半群，$\exp(\mathcal{S})$是它的指数，定义为$\mathcal{S}$中所有元素周期的最小公倍数.对于$\mathcal{S}$中元素的每个序列$T$（允许重复），令$σ(T) \in \mathcal{S}$表示$T$的所有项之和。定义$\mathcal{S}$的Davenport常数$D(\mathcal{S})$为最小正整数$d$，使得长度至少为$d$的$\mathcal{S}$上的每个序列$T$包含具有$σ(T')=σ(T)$的适当子序列$T'$，并定义了埃尔德什-金斯堡-Ziv定理常数$E(\mathcal{S})$是最小正整数$\ell$，使得长度至少为$\ell$的$\mathcal{S}$上的每个序列$T$包含具有$|T|-|T'|=\left\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})}\right\rceil\exp(\mathcal{S})$的子序列$T'$和$σ(T')=σ(T)$。当$\mathcal{S}$是有限交换群时，众所周知，$\left\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})}\right\rceil\exp(\mathcal{S})=|\mathcal{S}|$和$E(\mathcal{S…

2013-10-20math.ACmath.COmath.NT

Higher Degree Erdos-Ginzburg-Ziv Constants

We generalize the notion of Erdős-Ginzburg-Ziv constants -- along the same lines we generalized in earlier work the notion of Davenport constants -- to a ``higher degree" and obtain various lower and upper bounds. These bounds are sometimes exact as is the case for certain finite commutative rings of prime power cardinality. We also consider to what extent a theorem due independently to W.D.~Gao and the first author that relates these two parameters extends to this higher degree setting. Two simple examples that capture the essence of these higher degree Erdős-Ginzburg-Ziv constants are the fo…

2022-07-22math.ACmath.COmath.NT

$\mathbb{F}_q^n$上的高阶Erdős-Ginzburg-Ziv常数的界

Bounds on the Higher Degree Erdős-Ginzburg-Ziv Constants over $\mathbb{F}_q^n$

本研究针对有限域空间$\mathbb{F}_q^n$上的高阶Erdős-Ginzburg-Ziv常数，运用Lovász局部引理、Expurgation方法及Slice Rank等技术，推导其上下界，部分下界优于已有结果，并给出基于帽集问题相关结论的闭式上界。

2023-04-28math.COmath.NT

从函数在仿射线上的限制恢复其仿射线性

Recovering affine-linearity of functions from their restrictions to affine lines

本文研究从函数在仿射线上的限制恢复其全局仿射线性的条件，提出基于一般线和弱乘法B_h-集的定量界，并推广von Staudt结果至非双射情形。

2023-03-24math.ACmath.CO