The Erdos–Burgess constant of the multiplicative semigroup of a factor ring of Fq[x]
The Erdos–Burgess constant of the multiplicative semigroup of a factor ring of Fq[x]
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Fq[x] 因子环的乘法半群的 ErdosâBurgess 常数
DOI：
10.1142/s1793042119500015
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发表时间：
--
影响因子：
0.7
通讯作者：
Lizhen Zhang
中科院分区：
文献类型：
--
作者：
Haoli Wang;Jun Hao;Lizhen Zhang
设\(S\)是一个赋予了二元结合运算\(+\)的交换半群。\(S\)中的一个元素\(e\)如果满足\(e + e = e\)，则称\(e\)是幂等元。\(S\)的厄尔多斯 - 伯吉斯常数定义为满足以下条件的最小的\(\ell\in\mathbb{N}\cup\{\infty\}\)：来自\(S\)且长度为\(\ell\)的任何序列\(T\)都包含一个非空子序列，其各项之和是幂等元。设\(q\)是一个素数幂，并且设\(\mathbb{F}_q[x]\)是有限域\(\mathbb{F}_q\)上的多项式环。设\(R = \mathbb{F}_q[x]/K\)是\(\mathbb{F}_q[x]\)模任何理想\(K\)的商环。我们给出了环\(R\)的乘法半群的厄尔多斯 - 伯吉斯常数的一个精确下界，特别地，当\(K\)是\(\mathbb{F}_q[x]\)中一个素理想的幂或者两两不同的素理想的乘积时，我们确定了厄尔多斯 - 伯吉斯常数。
Let S be a commutative semigroup endowed with a binary associative operation +. An.20 element e of S is said to be idempotent if e + e = e. The Erd?os–Burgess constant of S.21 is defined as the smallest ∈ N ∪ {∞} such that any sequence T of terms from S and of.22 length contains a nonempty subsequence, the sum of whose terms is idempotent. Let.23 q be a prime power, and let Fq[x] be the polynomial ring over the finite field Fq. Let.24 R = Fq[x]K be a quotient ring of Fq[x] modulo any ideal K. We gave a sharp lower.25 bound of the Erd?os–Burgess constant of the multiplicative semigroup of the ring R, in.26 particular, we determined the Erd?os–Burgess constant in the case when K is the power.27 of a prime ideal or a product of pairwise distinct prime ideals in Fq[x].