If L is a first-order language, then an L-structure A is called quantifier-eliminable (QE) if every L-formula is equivalent in A to a formula without quantifiers.

The classification problem for QE groups and rings has received attention in work by Berline, Boffa, Cherlin, Feigner, Macintyre, Point, Rose, the present authors, and others. In [1], Berline and Cherlin reduced the problem for rings of prime characteristic p to that for nilrings, but also constructed countable QE nilrings of characteristic p. Likewise, in [3], we constructed countable QE nil-2 groups. Both results can be viewed as “nonstructure theorems”, in that they provide negative evidence for any attempt at classification. In the present paper we show that the situation is equally bad (or rich, depending on one's point of view) for commutative rings:

Theorem 1. For any odd prime p, there existcountable QE commutative nilrings of characteristic p.

This solves a problem posed in [1]. We remark that the examples we produce are uniformly locally finite, hence ℵ0-categorical. A more algebraic description is that each of our rings R is uniformly locally finite (in fact, R3 = 0) and homogeneous, in the sense that any isomorphism of finitely generated subrings extends to an automorphism of R.

Theorem 1 does not cover the case p = 2, and we show that for commutative rings this case is in fact exceptional:

Theorem 2. There exist exactly two nonisomorphic countably infinite QE commutative nilrings of characteristic 2.