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The {\\sl Erd\\H{o}s-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 $K$. 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