Radical classes of regular rings with Artinian primitive images

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Moreover, for each m, the class of strongly regular rings for which the m x m matrix ring is & is a radical class. This fact establishes a very important role for the radical classes of strongly regular rings, and accordingly a section is devoted to the latter. In the final section some attention is given to the question of closure of radical classes under direct products. It is probably safe to say that more is known about the radical classes which contain all nilpotent rings than about those which contain none. Certainly—with the (important) exception of the semisimple radical classes—the radical classes which have been most studied are towards the super nilpotent end of the spectrum. The present paper has three main aims: (1) to present some examples of (hereditary) subidempotent radical classes; (2) to examine radical theory a relatively tractable but nontrivial class of rings—the regular rings whose primitive homomorphic images are all artinian; (3) to say something about radical classes which are closed under directs. Investigations akin to objective (2), wherein radical theory is studied in microcosm, may be regarded as compromise substitutes for the (probably unrealistic) aim of describing all radical classes. The question of dirct product closure has been around for a long time. It is easy to see that a hereditary radical class with this closure property must be either supernilpotent of subidempotent, and we shall here be concerned with the latter, providing a few examples and counterexamples and answering, the negative, Richard Wiegandt's