交换单位环elementarily Equivalent到连通环限制积的公理
Axioms for Commutative Unital Rings elementarily Equivalent to Restricted Products of Connected Rings
摘要：
我们给公理的语言环增加了一个1元谓词符号$Fin(x)$与预期的解释，在布尔代数的幂等作为理想的有限元，即有限工会的原子。证明了满足这些公理的交换单位环初等等价于连通环的限制积。这是产品的\cite{elem-prod}结果的扩展。虽然\cite{elem-prod}中的结果给出了一个逆Feferman-Vaught定理的产品，我们的结果证明了同样的限制产品。本文用环的语言给出了数域上的Adeles环在数域上一致的一组完备公理。
英文摘要：
We give axioms in the language of rings augmented by a 1-ary predicate symbol $Fin(x)$ with intended interpretation in the Boolean algebra of idempotents as the ideal of finite elements, i.e. finite unions of atoms. We prove that any commutative unital ring satisfying these axioms is elementarily equivalent to a restricted product of connected rings. This is an extension of the results in \cite{elem-prod} for products. While the results in \cite{elem-prod} give a converse to the Feferman-Vaught theorem for products, our results prove the same for restricted products. We give a complete set of axioms in the language of rings for the ring of adeles of a number field, uniformly in the number field.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
