Journal of Symbolic Logic

Rings which Admit Elimination of Quantifiers

Bruce I. Rose

Abstract

We say that a ring admits elimination of quantifiers, if in the language of rings, $\{0, 1, +, \cdot\}$, 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: $\forall x \forall y \exists z (x = 0 \vee y = 0 \vee xzy \neq 0)$. 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 $\mathscr{A}$ be the class of finite fields. Let $\mathscr{B}$ be the class of $2 \times 2$ matrix rings over a field with a prime number of elements. Let $\mathscr{C}$ be the class of rings of the form $GF(p^n) \bigoplus GF(p^k)$ such that either $n = k$ or g.c.d. $(n, k) = 1$. Let $\mathscr{D}$ be the set of ordered pairs $(f, Q)$ where $Q$ is a finite set of primes and $f: Q \rightarrow \mathscr{A} \cup \mathscr{B} \cup \mathscr{C}$ such that the characteristic of the ring $f(q)$ is $q$. Finally, let $\mathscr{E}$ be the class of rings of the form $\bigoplus_{q \in Q}f(q)$ for some $(f, Q)$ in $\mathscr{D}$. 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 $\mathscr{E}$. 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 $\mathscr{E}$. 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.

Article information

Source
J. Symbolic Logic, Volume 43, Issue 1 (1978), 92-112.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183740110

Mathematical Reviews number (MathSciNet)
MR495121

Zentralblatt MATH identifier
0388.03010

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