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Regarding the construction of strict and base radical classes for the variety of associative rings, we show that "subring" and "accessible subring" are two of a number of substructures that can be used in constructing a radical class. The substructures we focus on are set properties that are transitive across accessible subrings. These substructures allow us to look beyond those that are related by the binary operations of the ring itself, to ones that may only be related by containment. The lower radical type construction defined by these substructures include those of Yu-lee Lee, McDougall and Stewart. [ABSTRACT FROM AUTHOR]