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Abstract
In this paper we investigate the radical properties of classes of rings constructed using a class pair (M 1 : M 2) of rings. Our theory encompasses the theory of radical pairs and allows the study of ring theoretical questions in terms of radicals. For instance, Hilbert's Nullstellensatz and an equivalent form of Koethe's conjecture can be stated in terms of class pairs.
We also determine conditions for (M 1 : M 2) to be a special radical. This enables us to show that many well-known classes of rings such as the Jacobson rings and many types of A-Jacobson rings Citation[12] form special radicals. Several open questions are also reformulated using our theory.
ACKNOWLEDGMENT
The authors wish to express their gratitude to the referee for the thorough reading and useful suggestions.