2020/07/17 by Jamshid Derakhshan, Angus Macintyre, Derakhshan, Jamshid +1
Computer Science · Mathematics · #Advanced Algebra and Logic #Commutative Algebra and Its Applications #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2007.09244
openalex publication_date 2020/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeelem-prod for products. While the results in \citeelem-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.