The purpose of this paper is
two-fold. First, we show that certain known properties of associative radicals
cannot be extended to the class of all rings. Secondly, we investigate classes of
(associative) rings closed under ideals and subdirect sums in relation to radical and
semisimple classes. Much of this is motivated by results obtained for torsion
theories in categories of modules by S. E. Dickson and J. P. Jans. We remark
that Dickson’s work was developed in the more general setting of Abelian
categories.