Corrigendum: “Rings which admit elimination of quantifiers”

1979 ◽  
Vol 44 (1) ◽  
pp. 109-110
Author(s):  
Bruce I. Rose
Keyword(s):  
Chain Condition ◽  
Prime Numbers ◽  
Matrix Rings ◽  
Zero Characteristic ◽  
Closed Fields ◽  
Semiprime Rings ◽  
Descending Chain Condition ◽  
Algebraically Closed Fields ◽  
Ordered Pairs

Matatyahu Rubin pointed out that the proof of Lemma 6.1 [2] works only for rings of prime or zero characteristic. This invalidates the characterization of semiprime rings with the descending chain condition on right or left ideals which admit elimination of quantifiers given in [2] and cited in the abstract [1]. Although the correct characterization is easy to derive, it is complex to state.Let be the class of finite fields. Let be the class of 2 × 2 matrix rings over a field with a prime number of elements. Let be the class of rings of the form GF(pn) ⊕ GF(pk) such that either n = k or g.c.d.(n, k) = 1 and p is a prime. Let ′ be the class of algebraically closed fields. Let P denote the set of all prime numbers together with zero. Let be the set of all ordered pairs (f, Q) where Q is a finite subset of P and f: Q → ⋃ ⋃ ⋃ such that the characteristic of the ring f(q) is q. Finally, let be the class of rings of the form ⊕q∈Qf(q) for some (f,Q) in .A corrected version of Theorem 6.2 [2] isTheorem 1. Let R be a ring with the descending chain condition on left or right ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if only if R belong to.

Rings which admit elimination of quantifiers

1978 ◽  
Vol 43 (1) ◽  
pp. 92-112 ◽  
Author(s):  
Bruce I. Rose
Keyword(s):  
Finite Field ◽  
Prime Ring ◽  
Chain Condition ◽  
Closed Field ◽  
Matrix Rings ◽  
Algebraically Closed Field ◽  
Finite Set ◽  
Descending Chain Condition ◽  
Infinite Center ◽  
Ordered Pairs

AbstractWe say that a ring admits elimination of quantifiers, if in the language of rings, {0, 1, +, ·}, the complete theory of R admits elimination of quantifiers.Theorem 1. Let D be a division ring. Then D admits elimination of quantifiers if and only if D is an algebraically closed or finite field.A ring is prime if it satisfies the sentence: ∀x∀y∃z (x =0 ∨ y = 0∨ xzy ≠ 0).Theorem 2. If R is a prime ring with an infinite center and R admits elimination of quantifiers, then R is an algebraically closed field.Let be the class of finite fields. Let be the class of 2 × 2 matrix rings over a field with a prime number of elements. Let be the class of rings of the form GF(pn)⊕GF(pk) such that either n = k or g.c.d. (n, k) = 1. Let be the set of ordered pairs (f, Q) where Q is a finite set of primes and such that the characteristic of the ring f(q) is q. Finally, let be the class of rings of the form ⊕q ∈ Qf(q), for some (f, Q) in .Theorem 3. Let R be a finite ring without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R belongs to.Theorem 4. Let R be a ring with the descending chain condition of left ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R is an algebraically closed field or R belongs to.In contrast to Theorems 2 and 4, we haveTheorem 5. If R is an atomless p-ring, then R is finite, commutative, has no nonzero trivial ideals and admits elimination of quantifiers, but is not prime and does not have the descending chain condition.We also generalize Theorems 1, 2 and 4 to alternative rings.

Presentations of skew fields. I. Existentially closed skew fields and the Nullstellensatz

1975 ◽  
Vol 77 (1) ◽  
pp. 7-19 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Algebraic Geometry ◽  
Matrix Ring ◽  
Commutative Rings ◽  
Commutative Case ◽  
Matrix Rings ◽  
Finite Dimensional ◽  
Existentially Closed ◽  
Closed Fields ◽  
Skew Fields ◽  
Algebraically Closed Fields

1. Introduction. The Nullstellensatz in commutative algebraic geometry may be described as a means of studying certain commutative rings (viz. affine algebras) by their homomorphisms into algebraically closed fields, and a number of attempts have been made to extend the result to the non-commutative case. In particular, Amitsur and Procesi have studied the case of general rings, with homomorphisms into matrix rings over commutative fields ((1), (2)) and Procesi has obtained more precise results for homomorphisms of PI-rings (11). Since a finite-dimensional division algebra can always be embedded in a matrix ring over a field, this includes the case of skew fields that are finite-dimensional over their centre, but it tells us nothing about general skew fields.

Model theory of strictly upper triangular matrix rings

1980 ◽  
Vol 45 (3) ◽  
pp. 455-463 ◽  
Author(s):  
William H. Wheeler
Keyword(s):  
Model Theory ◽  
Triangular Matrix ◽  
Closed Field ◽  
Matrix Rings ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Closed Fields ◽  
Upper Triangular Matrix ◽  
Algebraically Closed Fields ◽  
Model Completion

Two questions on rings of strictly upper triangular matrices arising from B. Rose's work [5] are answered in this paper. An n × n matrix (αi, j) is strictly upper triangular if αi, j = 0 whenever i ≥ j. The ring of strictly upper triangular n × n matrices with entries from a field F will be denoted by Sn(F). Throughout this paper n will be an integer greater than 2. B. Rose [5] has shown that the complete theory of Sn(F) for an algebraically closed field F is ℵ1categorical. The first main result of this paper is that the rings Sn(F) and Sn(K) for fields F and K are isomorphic or elementarily equivalent if and only if F and K are isomorphic or elementarily equivalent, respectively (Corollary 1.6 and Theorem 2.2). This result shortens the proof of B. Rose's categoricity theorem [5, Theorem 7] by avoiding the co-stability considerations; furthermore, this result yields a proof of the converse of this categoricity theorem. The second main result is that the theory of rings of strictly upper triangular n × n matrices over algebraically closed fields is the model-completion of the theory of rings of strictly upper triangular n × n matrices over arbitrary fields (Theorem 2.5). This answers affirmatively the two conjectures at the end of [5].

The ℵ1-categoricity of strictly upper triangular matrix rings over algebraically closed fields

1978 ◽  
Vol 43 (2) ◽  
pp. 250-259 ◽  
Author(s):  
Bruce I. Rose
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Closed Field ◽  
Matrix Rings ◽  
Algebraically Closed Field ◽  
Closed Fields ◽  
Upper Triangular Matrix ◽  
Algebraically Closed Fields

AbstractLet 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 ⊨ φ if and only if R φ σ(φ).Theorem. The theory of a strictly upper triangular n × n matrix ring over an algebraically closed field is ℵ1-categorical.

scholarly journals Equationally Compact Artinian Rings

1973 ◽  
Vol 25 (2) ◽  
pp. 273-283 ◽  
Author(s):  
David K. Haley
Keyword(s):  
Artinian Ring ◽  
Chain Condition ◽  
Finite Rings ◽  
Direct Sums ◽  
Artinian Rings ◽  
Chain Conditions ◽  
Descending Chain Condition ◽  
Equational Compactness ◽  
Theoretical Results

By a Noetherian (Artinian) ring=(R;+ , —, 0, ·) we mean an associative ring satisfying the ascending (descending) chain condition on left ideals. An arbitrary ringis said to beequationally compactif every system of ring polynomial equations with constants inis simultaneously solvable inprovided every finite subset is. (The reader is referred to [2; 8; 13; 14] for terminology and relevant results on equational compactness, and to [4] for unreferenced ring-theoretical results.) In this report a characterization of equationally compact Artinian rings is given - roughly speaking, these are the finite direct sums of finite rings and Prüfer groups; as consequences it is shown that an equationally compact ring satisfying both chain conditions is always finite, as is any Artinian ring which is a compact topological ring.

scholarly journals Grouplike minimal sets in ACFA AND in TA

2010 ◽  
Vol 75 (4) ◽  
pp. 1462-1488
Author(s):  
Alice Medvedev
Keyword(s):  
Algebraic Geometry ◽  
Algebraic Curve ◽  
Model Companion ◽  
Full Characterization ◽  
Closed Fields ◽  
Transcendental Theory ◽  
Phd Thesis ◽  
Algebraically Closed Fields ◽  
Injective Endomorphism

AbstractThis paper began as a generalization of a part of the author's PhD thesis about ACFA and ended up with a characterization of groups definable in TA. The thesis concerns minimal formulae of the form x Є A ∧ σ(x) = f(x) for an algebraic curve A and a dominant rational function f: A → σ(A). These are shown to be uniform in the Zilber trichotomy, and the pairs (A, f) that fall into each of the three cases are characterized. These characterizations are definable in families. This paper covers approximately half of the thesis, namely those parts of it which can be made purely model-theoretic by moving from ACFA, the model companion of the class of algebraically closed fields with an endomorphism, to TA, the model companion of the class of models of an arbitrary totally-transcendental theory T with an injective endomorphism, if this model-companion exists. A TA analog of the characterization of groups definable in ACFA is obtained in the process. The full characterization of the cases of the Zilber trichotomy in the thesis is obtained from these intermediate results with heavy use of algebraic geometry.

Une Caractérisation des Polynômes Prenant des Valeurs Entières Sur Tous les Nombres Premiers

1996 ◽  
Vol 39 (4) ◽  
pp. 402-407 ◽  
Author(s):  
Jean-Luc Chabert
Keyword(s):  
Prime Numbers

AbstractWe give a characterization of polynomials with rational coefficients which take integral values on the prime numbers: to test a polynomial of degree n, it is enough to consider its values on the integers from 1 to 2n —1.

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