Let K be an infinite Galois extension of the rationals such that every finite subextension has odd degree over the rationals and its prime ideals dividing 2 are unramified. We show that its ring of integers is first-order definable in K. As an application we prove that equation image together with all its Galois subextensions are undecidable, where Δ is the set of all the prime integers which are congruent to −1 modulo 4.

One of the main theorems of the paper states the following. Let R-K-M be finite extensions of a rational one variable function field R over a finite field of constants. Let S be a finite set of valuations of K. Then the ring of elements of K having no poles outside S has a Diophantine definition over its integral closure in M.

We consider the problem of constructing first-order definitions in the language of rings of holomorphy rings of one-variable function fields of characteristic 0 in their integral closures in finite extensions of their fraction fields and in bigger holomorphy subrings of their fraction fields. This line of questions is motivated by similar existential definability results over global fields and related questions of Diophantine decidability.

This paper provides the first examples of rings of algebraic numbers containing the rings of algebraic integers of the infinite algebraic extensions of where Hilbert's Tenth Problem is undecidable.

We show that elliptic curves whose Mordell–Weil groups are finitely generated over some infinite extensions of ${\mathbb {Q}}$ , can be used to show the Diophantine undecidability of the rings of integers and bigger rings contained in some infinite extensions of rational numbers.

LetKbe an algebraic number field andIKthe ring of algebraic integers inK. *Kand *IKdenote enlargements ofKandIKrespectively. LetxЄ *K–K. In this paper, we are concerned with algebraic extensions ofKwithin *K. For eachxЄ *K–Kand each natural numberd, YKis defined to be the number of algebraic extensions ofKof degreedwithin *K.xЄ *K–Kis called a Hilbertian element ifYK= 0 for alldЄ N,d> 1; in other words,Khas no algebraic extension within *K. In their paper [2], P. C. Gilmore and A. Robinson (...) proved that the existence of a Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural numberω, 2ωPωand 2ω are Hilbertian elements in*Q, where pωis theωth prime number. (shrink)

We give necessary conditions for a set to be definable by a formula with a universal quantifier and an existential quantifier over algebraic integer rings or algebraic number fields. From these necessary conditions we obtain some undefinability results. For example, N is not definable by such a formula over Z. This extends a previous result of R. M. Robinson.

When is an ideal of a ring radical or prime? By examining its generators, one may in many cases definably and uniformly test the ideal’s properties. We seek to establish such definable formulas in rings of p-adic power series, such as $\mathbb Q_{p}\langle X\rangle $, $\mathbb Z_{p}\langle X\rangle $, and related rings of power series over more general valuation rings and their fraction fields. We obtain a definable, uniform test for radicality, and, in the one-dimensional case, for primality. This (...) builds upon the techniques stemming from the proof of the quantifier elimination results for the analytic theory of the p-adic integers by Denef and van den Dries, and the linear algebra methods of Hermann and Seidenberg.prepared by Madeline G. Barnicle.E-mail: [email protected]: https://escholarship.org/uc/item/6t02q9s4. (shrink)

The archetypal Rumely domain is the ring \widetildeZ of algebraic integers. Its constructible Boolean algebra is atomless. We study here the opposite situation: Rumely domains whose constructible Boolean algebra is atomic. Recursive models (which are rings of algebraic numbers) are proposed; effective model-completeness and decidability of the corresponding theory are proved.

The Post, axled and Łukasiewicz–Moisil algebras are important lattices studied in algebraic logic. In this paper, we investigate a useful interpretation between these algebras and some rings. We give a term equivalence between Post algebras of order |$p$| and |$p$|-rings, |$p$| prime and lift this result to the axled Łukasiewicz–Moisil algebra |$L \cong B_s \times P$| and the ring |$\prod ^s F_2 \times \prod ^l F_p$|⁠, where |$B_s$| is a Boolean algebra of order |$2^s$|⁠, |$P$| a |$p$|-valued Post (...) algebra of order |$p^l$| and |$F_p$| is the prime field of order |$p$|⁠. (shrink)

Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, (...) or an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained in [Formula: see text] and naturally isomorphic to [Formula: see text], such that the ring of functions [Formula: see text] which take values in [Formula: see text] is definable in [Formula: see text]. (shrink)

In this work we introduce a class of commutative rings whose defining condition is that its lattice of ideals, augmented with the ideal product, the semi-ring of ideals, is isomorphic to an MV-algebra. This class of rings coincides with the class of commutative rings which are direct sums of local Artinian chain rings with unit.

In a number of publications, John Earman has advocated a tertium quid to the usual dichotomy between substantivalism and relationism concerning the nature of spacetime. The idea is that the structure common to the members of an equivalence class of substantival models is captured by a Leibniz algebra which can then be taken to directly characterize the intrinsic reality only indirectly represented by the substantival models. An alleged virtue of this is that, while a substantival interpretation of spacetime theories falls (...) prey to radical local indeterminism, the Leibniz algebras do not. I argue that the program of Leibniz algebras is subject to radical local indeterminism to the same extent as substantivalism. In fact, for the category of topological spaces of interest in spacetime physics, the program is equivalent to the original spacetime approach. Moreover, the motivation for the program--that isomorphic substantival models should be regarded as representing the same physical situation--is misguided. (shrink)

We prove that the theory of all sentences in the language of rings which are true in for almost all is decidable. Here is the field of all algebraic numbers; is the ring of all algebraic integers; is the absolute Galois group of ; for each , is the fixed field of σ1,…,σe in ; and the clause ‘almost all’ is used in the sense of the Haar measure of.

The fields K) of generalized power series with coefficients in a field K and exponents in an additive abelian ordered group G play an important role in the study of real closed fields. The subrings K) consisting of series with non-positive exponents find applications in the study of models of weak axioms for arithmetic. Berarducci showed that the ideal JK) generated by the monomials with negative exponents is prime when is the additive group of the reals, and asked whether the (...) same holds for any G. We prove that this is the case and that in the quotient ring K)/J, each element admits at least one factorization into irreducibles. (shrink)

We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with signature $$. 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.

We study relationships between certain algebraic properties of groups and rings definable in a first order structure or *-closed in a compact G-space. As a consequence, we obtain a few structural results about ω-categorical rings as well as about small, nm-stable compact G-rings, and we also obtain surprising relationships between some conjectures concerning small profinite groups.

ABSTRACT In this paper we shall prove that l-rings are categorally equivalent to the MV*-algebras, a subcategory of perfect MV-algebras. We shall use this equivalence in order to characterize l-rings as quotients of certain semirings of matrices over MV*-algebras. We shall establish a relation between l-ideals in l-rings and some ideals in MV*-algebras. This edlows us to study the MV* f-algebras, a subclass of the MV*-algebras corresponding to the f-rings.

In this paper we consider properties, related to model-completeness, of the theory of integrally closed commutative regular rings. We obtain the main theorem claiming that in a Boolean algebra B, the truth of a prenex Σn-formula whose parameters ai partition B, can be determined by finitely many conditions built from the first entry of Tarski invariant T(ai)'s, n-characteristic D(n, ai)'s and the quantities S(ai, l) and S'(ai, l) for $l < n$. Then we derive two important theorems. One claims that (...) for any Boolean algebras A and B, an embedding of A into B preserving D(n, a) for all a ∈ A is a Σn-extension. The other claims that the theory of n-separable Boolean algebras admits elimination of quantifiers in a simple definitional extension of the language of Boolean algebras. Finally we translate these results into the language of commutative regular rings. (shrink)

We show that R.J. Thompson’s groupsFandTare bi-interpretable with the ring of the integers. From a result by A. Khélif, these groups are quasi-finitely axiomatizable and prime. So, the groupTprovides an example of a simple group which is quasi-finitely axiomatizable and prime. This answers questions posed by T. Altınel and A. Muranov in [2], and by A. Nies in [12].

We consider ω n -automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length ω n for some integer n ≥ 1. We show that all these structures are ω-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for ω 2 -automatic (resp. ω n -automatic for n > 2) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is (...) not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for ω n -automatic boolean algebras, n ≥ 2, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a ${\mathrm{\Sigma }}_{2}^{1}-\mathrm{s}\mathrm{e}\mathrm{t}$ nor a ${\mathrm{\Pi }}_{2}^{1}-\mathrm{s}\mathrm{e}\mathrm{t}$ . We obtain that there exist infinitely many ω n -automatic, hence also ω-tree-automatic, atomless boolean algebras $\mathcal{B}_{n}$ , n ≥ 1, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [14]. (shrink)

Let be the field of p‐adic numbers in the language of rings. In this paper we consider the theory of expanded by two predicates interpreted by multiplicative subgroups and where are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic if α and β have positive p‐adic valuation. If either α or β has zero valuation we show that the theory of has the NIP (“negation of the independence property”) and therefore does not interpret Peano arithmetic. (...) In that case we also prove that the theory is decidable if and only if the theory of is decidable. (shrink)

An explicit categorical equivalence is defined between a proper subvariety of the class of \-algebras, as defined by Di Nola and Dvurečenskij, to be called \-algebras, and the category of semi-low \-rings. This categorical representation is done using the prime spectrum of the \-algebras, through the equivalence between \-algebras and \-groups established by Mundici, from the perspective of the Dubuc–Poveda approach, that extends the construction defined by Chang on chains. As a particular case, semi-low \-rings associated to Boolean algebras are (...) characterized. (shrink)

An explicit categorical equivalence is defined between a proper subvariety of the class of \-algebras, as defined by Di Nola and Dvurečenskij, to be called \-algebras, and the category of semi-low \-rings. This categorical representation is done using the prime spectrum of the \-algebras, through the equivalence between \-algebras and \-groups established by Mundici, from the perspective of the Dubuc–Poveda approach, that extends the construction defined by Chang on chains. As a particular case, semi-low \-rings associated to Boolean algebras are (...) characterized. (shrink)

Let n ≥ 3. The following theorems are proved. Theorem. The theory of the class of strictly upper triangular n × n matrix rings over fields is finitely axiomatizable. Theorem. If R is a strictly upper triangular n × n matrix ring over a field K, then there is a recursive map σ from sentences in the language of rings with constants for K into sentences in the language of rings with constants for R such that $K \vDash \varphi$ (...) if and only if $R \vDash \sigma$. Theorem. The theory of a strictly upper triangular n × n matrix ring over an algebraically closed field is ℵ 1 -categorical. (shrink)

Given a theory T of a polynomially bounded o-minimal expansion R of R¯=⟨R,+,.,0,1,<⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\mathbb{R}} = \langle\mathbb{R}, +,., 0, 1, < \rangle}$$\end{document} with field of exponents Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q}}$$\end{document}, we introduce a theory T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{T}}$$\end{document} whose models are expansions of dense pairs of models of T by a discrete multiplicative group. We prove that T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...) \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{T}}$$\end{document} is complete and admits quantifier elimination when predicates are added for certain existential formulas. In particular, if T = RCF then T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{T}}$$\end{document} axiomatises ⟨R¯,Ralg,2Z⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle\bar{\mathbb{R}}, \mathbb{R}_{alg}, 2^{\mathbb{Z}}\rangle}$$\end{document}, where Ralg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}_{alg}}$$\end{document} denotes the real algebraic numbers. We describe types and definable sets in our models and prove that T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{T}}$$\end{document} is dependent. (shrink)

Let R be a real closed field, and X⊆Rm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\subseteq R^m}$$\end{document} semi-algebraic and 1-dimensional. We consider complete first-order theories of modules over the ring of continuous semi-algebraic functions X→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\to R}$$\end{document} definable with parameters in R. As a tool we introduce -piecewise vector bundles on X and show that the category of piecewise vector bundles on X is equivalent to the category (...) of syzygies of finitely generated submodules of free modules. We give an explicit method to determine the Baur–Monk invariants of free modules in terms of pre-piecewise vector bundles. When R is a recursive real closed field this yields the decidability of the theory of free modules. Where it makes sense, we address the same questions for continuous definable functions in o-minimal expansions of a real closed field. From the free module case we are able to deduce generalisations of some results to arbitrary modules over the ring. We present a geometrically motivated quantifier elimination result down to the level of positive primitive formulae with a certain block decomposition of the matrix of coefficients. (shrink)

Let R be a real closed field. An integer part I for R is a discretely ordered subring such that for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r \in R}$$\end{document}, there exists an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${i \in I}$$\end{document} so that i ≤ r < i + 1. Mourgues and Ressayre (J Symb Logic 58:641–647, 1993) showed that every real closed field has an integer part. The procedure of (...) Mourgues and Ressayre appears to be quite complicated. We would like to know whether there is a simple procedure, yielding an integer part that is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2(R)}$$\end{document} —limit computable relative to R. We show that there is a maximal Z-ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I \subseteq R}$$\end{document} which is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2(R)}$$\end{document}. However, this I may not be an integer part for R. By a result of Wilkie (Logic Colloquium ’77), any Z-ring can be extended to an integer part for some real closed field. Using Wilkie’s ideas, we produce a real closed field R with a Z-ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I \subseteq R}$$\end{document} such that I does not extend to an integer part for R. For a computable real closed field, we do not know whether there must be an integer part in the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2}$$\end{document}. We know that certain subclasses of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2}$$\end{document} are not sufficient. We show that for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \in \omega}$$\end{document}, there is a computable real closed field with no n-c.e. integer part. In fact, there is a computable real closed field with no n-c.e. integer part for any n. (shrink)

This paper surveys the recent developments in the area that grew out of attempts to solve an analog of Hilbert's Tenth Problem for the field of rational numbers and the rings of integers of number fields. It is based on a plenary talk the author gave at the annual North American meeting of ASL at the University of Notre Dame in May of 2009.

We present numerous natural algebraic examples without the so-called Canonical Base Property (CBP). We prove that every commutative unitary ring of finite Morley rank without finite-index proper ideals satisfies the CBP if and only if it is a field, a ring of positive characteristic or a finite direct product of these. In addition, we construct a CM-trivial commutative local ring with a finite residue field without the CBP. Furthermore, we also show that finite-dimensional non-associative algebras over (...) an algebraically closed field of characteristic [Formula: see text] give rise to triangular rings without the CBP. This also applies to Baudisch’s [Formula: see text]-step nilpotent Lie algebras, which yields the existence of a [Formula: see text]-step nilpotent group of finite Morley rank whose theory, in the pure language of groups, is CM-trivial and does not satisfy the CBP. (shrink)

This book is intended for mathematicians. Its origins lie in a course of lectures given by an algebraist to a class which had just completed a sub stantial course on abstract algebra. Consequently, our treatment ofthe sub ject is algebraic. Although we assurne a reasonable level of sophistication in algebra, the text requires little more than the basic notions of group, ring, module, etc. A more detailed knowledge of algebra is required for some of . the exercises. We (...) also assurne a familiarity with the main ideas of set theory, including cardinal numbers and Zorn's Lemma. In this book, we carry out a mathematical study of the logic used in mathematics. We do this by constructing a mathematical model oflogic and applying mathematics to analyse the properties of the model. We therefore regard all our existing knowledge of mathematics as being applicable to the analysis of the model, and in particular we accept set theory as part of the meta-Ianguage. We are not attempting to construct a foundation on which all mathematics is to be based-rather, any conclusions to be drawn about the foundations of mathematics co me only by analogy with the model, and are to be regarded in much the same way as the conclusions drawn from any scientific theory. (shrink)