A note on Mal'tsev-Neumann products of radical classes

A radical class 𝑅 of rings is elementary if it contains precisely those rings whose singly generated subrings are in 𝑅. Many examples of elementary radical classes are presented, and all those which are either contained in the Jacobson radical class or disjoint from it are described. There is a discussion of Mal'tsev products of radical classes in general, in which it is shown, among other things, that a product of elementary radical classes need not be a radical class, and if it is, it need not be elementary.