We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are (commutative unital) reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with signature $(0,1,+,\cdot )$. In the characteristic zero case, the claim implies that the full theory is undecidable, for rings of the referred type. This extends a series of results by Raphael Robinson, holding for certain polynomial integral domains, to a more general class.

中文翻译：

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约简不可分解多项式环中整数的统一可定义性

我们证明了多项式环内素数子环的一阶可定义性，其系数环（可交换单位）减少且不可分解。这是通过带有签名的戒指语言中的统一公式来实现的$(0,1,+,\cdot )$. 在特征零的情况下，该声明意味着对于所指类型的环，完整的理论是不可判定的。这将 Raphael Robinson 对某些多项式积分域持有的一系列结果扩展到更一般的类别。