The results of this paper
combine two areas of abstract mathematics: the theory of radicals for associative or
alternative rings, and model theory for universal algebras. The main theorem
provides necessary and sufficient conditions on the radical of any product ring and
on the radical of any ultraproduct ring in order that the radical class and
its corresponding semisimple class be finitely axiomatic (elementary). As
a corollary, it follows that if a radical class of rings and its corresponding
semisimple class are axiomatic, then they are both finitely axiomatic. In
addition, several subsidiary results are given, and unanswered questions
posed.
