In:
Journal of Symbolic Logic, Cambridge University Press (CUP), Vol. 43, No. 1 ( 1978-03), p. 92-112
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). T heorem 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(p n )⊕GF(p k ) 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 ∈ Q f(q) , for some ( f, Q ) in . T heorem 3. Let R be a finite ring without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R belongs to . T heorem 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 T heorem 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 of Medium:
Online Resource
ISSN:
0022-4812
,
1943-5886
Language:
English
Publisher:
Cambridge University Press (CUP)
Publication Date:
1978
detail.hit.zdb_id:
2010607-5
SSG:
5,1
SSG:
17,1
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