Corrigendum: “Rings which admit elimination of quantifiers”

Extract

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(p^k) 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] is

Theorem 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.