We work in the language of rings augmented by a 1-ary predicate symbol Fin(x) with intended interpretation in a ring as "x is a finite union of atoms" in the Boolean algebra of idempotents of the ring. We exhibit a set of axioms in this language, and prove that any commutative unital ring R satisfying these axioms is elementarily equivalent to a restricted product, over the set of atoms e of R, of connected rings Re. Each connected ring Re is the localization of Rat the set of powers of e. This proves a Feferman-Vaught theorem for rings and a converse to the Feferman-Vaught theorem for restricted products of rings. The most important application is to axioms for rings closely resembling adele rings over number fields. Our axioms are inspired by the axioms of D'Aquino and Macintyre for products, and our results are an exact analogue of their results on products which have intriguing applications to nonstandard models of first-order Peano arithmetic.

We prove that the product of any family of pseudofinite structures is pseudofinite using the fundamental work on products of first-order structures due to Feferman and Vaught (1959), exploiting the underlying combinatorics.

We prove that the class of all the rings Z/mZ for all m > 1 is decidable. This gives a positive solution to a problem of Ax asked in his celebrated 1968 paper on the elementary theory of finite fields [1, Problem 5, p. 270]. In our proof, we reduce the problem to the decidability of the ring of adeles A(Q) of Q.

The classical work of Feferman Vaught gives a powerful, constructive analysis of definability in (generalized) product structures, and certain associated enriched Boolean structures. %structures in terms of definability in the component structures. Here, by closely related methods, but in the special setting of commutative unital rings, we obtain a kind of converse allowing us to determine in interesting cases, when a commutative unital R is elementarily equivalent to a nontrivial product of a family of commutative unital rings R_i. We use this in the model theoretic analysis of residue rings of models of Peano Arithmetic.

We study the model theory of the ring of adeles of a number field. We obtain quantifier elimination results in the language of rings and some enrichments. We given consequences for definable subsets of the adeles, and their measures.

In the second author gave a systematic analysis of definability and decidability for rings ℳ/pℳ, where ℳ is a model of Peano Arithmetic and p is a prime in ℳ. In the present paper we extend those results to the more difficult case of ℳ/p^kℳ, where ℳ is a model of Peano Arithmetic, p is a prime in ℳ, and k>1. In work of Ax on finite fields was used, here we use in addition work of Ax on ultraproduct of p-adics.

We use the classical Ax-Kochen-Ershov analysis of the model theory of Henselian fields to bring out some model-theoretical aspects of the structure sheaf of the spectrum of Z^ and the ring of finite adeles over Q. We show that various structures associated to a prime ideal, such as quotients and localizations, are well understood model-theoretically, and they are closely connected to ultrafilters on the set of standard primes.

We use the classical Ax-Kochen-Ershov analysis of the model theory of Henselian fields to bring out some model-theoretical aspects of the structure sheaf of the spectrum of Ẑ and the ring of finite adèles over Q. We show that various structures associated to a prime ideal, such as quotients and localizations, are well understood model-theoretically, and they are closely connected to ultrafilters on the set of standard primes. 1. Notation and Basic Notions We will use the following notation: P = the set of prime numbers; Zp = the ring of p-adic integers; Qp = the field of p-adic numbers; μp = maximal ideal of Zp, and

We study elementary equivalence of adele rings and decidability for adele rings of general number fields. We prove that elementary equivalence of adele rings implies isomorphism of the adele rings.

We consider abelian groups G elementarily equivalent to Z, the additive group of integers. We give an exact classification of the isomorphism types of those G which admit a pure embedding of Z, by using homological algebra and adelic ideas to be found in Tate's thesis (and given an elementary presentation in an expository paper of Keith Conrad).

Our understanding of the first-order theory of the class of all local rings Z/p(n) Z as p and n vary comes from the Ax-Kochen-Ershov analysis of the rings of p-adic integers. This analysis does not directly produce axioms. In this paper we give fairly explicit axioms for the class.

We give a unified treatment of the model theory of various enrichments of infinite atomic Boolean algebras, with special attention to quantifier eliminations, complete axiomatizations and decidability. Our main enrichment is by a predicate for the ideal of finite sets and predicates for congruence conditions on the cardinalities of finite sets, but we also give new proofs of some classical results. We then classify and compare the expressive power of the enriched theories.

I show that all Zilber's countable strong exponential fields are computable exponential fields.

We continue the research programme of comparing the complex exponential with Zilberś exponential. For the latter, we prove, using diophantine geometry, various properties about zero sets of exponential functions, proved for $\mathbb{C}$ using analytic function theory, for example, the Identity Theorem.

We prove that the theory of a Henselian valued field of characteristic zero, with finite ramification, and whose value group is a $Z$-group, is model-complete in the language of rings if the theory of its residue field is model-complete in the language of rings. We apply this to prove that every infinite algebraic extension of the field of $p$-adic numbers $\Bbb Q_p$ with finite ramification is model-complete in the language of rings. For this, we give a necessary and sufficient condition for model-completeness of the theory of a perfect pseudo-algebraically closed field with pro-cyclic absolute Galois group.