论文标题
通用联合环的公理基本等同于连接环的受限产物
Axioms for Commutative Unital Rings elementarily Equivalent to Restricted Products of Connected Rings
论文作者
论文摘要
我们以1个谓词符号$ fin（x）$增强的戒指的语言给出公理，并在偶像群体的布尔代数中具有预期的解释，这是有限元素的理想，即原子有限的工会。我们证明，满足这些公理的任何换向的联合环在基本上等同于连接环的受限产物。这是\ cite {elem-prod}的结果的扩展。虽然\ cite {elem-prod}中的结果与feferman-wived产品的定理进行了交谈，但我们的结果证明对限制产品的结果相同。我们在数字字段中均匀地提供了一个数字字段的环形戒指的戒指语言的完整公理。
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.
