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

关于模Erdens-Burgess常数

On the modular Erdős-Burgess constant

令$n$为正整数。对于任何整数$a$，如果$a^2\equiv a\pmod n$，我们说$a$是模$n$幂等的。$n$模Erdens-Burgess常数是最小的正整数$\ell$，使得任何$\ell$整数都包含一个或多个整数，其乘积是模$n$幂等的。给出了$n$ -模Erdens-Burgess常数的一个精确下界，特别是当$n$是素数幂或两两不同素数的乘积时，给出了$n$ -模Erdens-Burgess常数的一个精确下界.

Existence of Erdős-Burgess constant in commutative rings

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 f…

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…

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…

Erdens-Burgess常数和Davenport常数之间的更强联系

A stronger connection between the Erdős-Burgess and Davenport constants

半群$S$的Erdens-Burgess常数是最小的正整数$k$，使得在$S$上长度为$k$的任何序列包含一个非空子序列，其元素乘以$S$的幂等元。在$S$是$\mathbb{Z}/n\mathbb{Z}$的乘法半群的情形下，我们证明了一个猜想，当$n$至多有两个素因子时，它将$S$的Erdess-Burgess常数与$(\mathbb{Z}/n\mathbb{Z})^{\times}$的Davenport常数联系起来.我们还讨论了我们的技术扩展到其他环。

关于$\mathbb{F}_q[x]$的因子环的乘法半群的Erdens-Burgess常数

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

设$\mathcal{S}$是具有二元结合运算$+$的交换半群. $\mathcal{S}$的元素$e$称为幂等的，如果$e+e=e$。$\mathcal{S}$的{ \sl Erdens-Burgess常数}被定义为最小的$\ell\in \mathbb{N}\cup \{\infty\}$，使得来自$S$且长度为$\ell$的项的任何序列$T$包含其项的和是幂等的非空子序列。设$q$是素数幂，$\F_q[x]$是有限域$\F_q$上的多项式环.设$R=\F_q[x]\diagup K$是$\F_q[x]$模任意理想$K$的商环.给出了环$R$的乘法半群的Erdens-Burgess常数的一个精确下界，特别是当$K$是$\F_q[x]$中素理想或两两不同素理想的乘积的幂时，给出了Erdens-Burgess常数的一个精确下界.