To the Abstract Theory of Radicals: A Contribution from Near-Rings

Abstract

This chapter discusses the abstract theory of radicals. The concept of radical classes has proved to be so wide that it not only includes the classical radicals of associative rings but also radicals of a completely opposite character where the radical contains the idempotent ideals. The abstract theory of radicals has as foundation the category of groups and especially the category of associative rings. It has grown into an autonomous subject of research investigations. Although a number of the results of the associative ring case can be formulated and proved in much wider contexts, there are several results that depend on the underlying axioms and consequently, these results cannot be generalized using only homological tools. The variety of near-rings has much to contribute to the abstract theory of radicals. From an axiomatic viewpoint, there is not much difference between a ring and a near-ring. Near-ringers like to distinguish their objects of study from rings by proclaiming that ring theory is the tool for linear algebra and near-ring theory the tool for non-linear algebra.