论文标题
在交换环中存在Erdős-Burgess常数
Existence of Erdős-Burgess constant in commutative rings
论文作者
论文摘要
令$ r $为可交换的统一戒指。 $ r $中的iDempotent是$ e $ in $ e^2 = e $的元素$ e \。与环$ r $相关的erdős-burgess常数是最小的正整数$ \ ell $（如果存在），以至于对于任何给定的$ \ ell $ emlements（不一定是不同的）$ r $的元素（例如$ a_1，\ ldots，\ ldots，a _ {\ ell} \ {1,2，\ ldots，\ ell \} $带有$ \ prod \ limits_ {j} a_j $ as diadempotent。在本文中，我们证明，除了具有非常特殊形式的无限通勤戒指外，如果且只有$ r $是有限的，则存在ring $ r $的erdős-burgess常数。
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.
