Closure properties in radical theory

Abstract

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.