Finite QE rings in characteristic \(p^ 2\) (Q762141)

This paper deals with the classification of quantifier-eliminable (QE) rings. Previous work has reduced the classification to that of the Jacobson radical J of QE rings of char. \(p^ n\) [for a summary see: \textit{Ch. Berline} and \textit{G. Cherlin}, J. Symb. Logic 48, 140-162 (1983; Zbl 0524.03016)]. J is a nilring, QE for the language \((0,+,.,p)\) and it is known that, even in the commutative case, there are \(2^{\aleph_ 0}\) countable such rings [\textit{D. Saracino} and \textit{C. Wood}, J. Symb. Logic 49, 644-651 (1984)]. The present paper provides a complete classification of the finite rings which occur as the Jacobson radical of a finite QE \((=finite\) homogeneous) ring of char. \(p^ 2\), p odd. Some results have relevance beyond the \(p^ 2\)-case and the authors announce a subsequent paper on the \(p^ n\)-case.