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Rings which admit elimination of quantifiers

Published online by Cambridge University Press:  12 March 2014

Bruce I. Rose
Bruce I. Rose*
Affiliation:
University of Notre Dame, Notre Dame, Indiana 46556
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Abstract

We 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 have

Theorem 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 1 , March 1978 , pp. 92 - 112
Copyright
Copyright © Association for Symbolic Logic 1978

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References

REFERENCES

[1]Baldwin, J. T. and Lachlan, A. H., On strongly minimal sets, this Journal, vol. 36 (1971), pp. 79–96.Google Scholar
[2]Baldwin, J. T. and Rose, B. I., ℵ0-categoricity and stability of rings, Journal of Algebra, vol. 45 (1977), pp. 1–16.CrossRefGoogle Scholar
[3]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1972.Google Scholar
[4]Cherlin, G. L. and Reineke, J., Categoricity and stability of commutative rings, Annals of Mathematical Logic, vol. 9 (1976), pp. 367–400.CrossRefGoogle Scholar
[5]Felgner, U., Ringe, deren Theorien ℵ1-categorich sind, Fundamenta Mathematicae, vol. 72 (1975), pp. 331–346.CrossRefGoogle Scholar
[6]Foster, A. L., p-rings and their Boolean-vector representation, Acta Mathematica, vol. 84 (1951), pp. 231–261.CrossRefGoogle Scholar
[7]Halmos, P. R., Lectures on Boolean algebras, Van Nostrand, Princeton, 1963.Google Scholar
[8]Herstein, I. N., Noncommutative rings, Mathematical Association of America Carus Monograph Series, vol. 15, The American Mathematical Society, Providence, R.I., 1968.Google Scholar
[9]Jacobson, N., Structure of rings, 2nd edition, American Mathematical Society Colloquium Publications, vol. 37, The American Mathematical Society, Providence, R.I., 1970.Google Scholar
[10]Kaplansky, I., Fields and rings, University of Chicago Press, Chicago, Illinois, 1969.Google Scholar
[11]Macintyre, A., On ω1-categorical theories of fields, Fundamenta Mathematicae, vol. 71 (1971), pp. 1–25.CrossRefGoogle Scholar
[12]Rose, B. I., Model theory of alternative rings, Ph.D. Thesis, University of Chicago, 1976.Google Scholar
[13]Sabbagh, G., Embedding problems for modules and rings with application to model-companions, Journal of Algebra, vol. 18 (1971), pp. 390–403.CrossRefGoogle Scholar
[14]Sacks, G. E., Saturated model theory, Benjamin, Reading, Massachusetts, 1972.Google Scholar
[15]Schafer, R. D., An introduction to nonassociative algebras, Academic Press, New York, 1966.Google Scholar
[16]Slater, M., Strongly prime alternative rings (to appear).Google Scholar
[17]Tarski, A., Contributions to the theory of models, Proceedings of the Royal Academy of Sciences, Amsterdam, series A, vol. 57 (1954), pp. 572–581, pp. 582–588 and vol. 58 (1955), pp. 56–64.Google Scholar