Introduction

The kind of theory developed here follows a pattern which is discernible in other such related theories. One starts with some given class ∑ of modules, and then defines the radical corresponding to ∑ to be the intersection of the annihilator ideals of all the modules in ∑.

Set theory is applicable mostly to sets, but possibly not always to the larger entities called classes, like the class of all (one sided) simple modules ∑ used in this chapter. However, module isomorphism is an equivalence relation ∼ on ∑. The equivalence classes modulo this relation do form a set, the set ∑/ ∼ of all isomorphism classes of simple modules. By the axiom of choice we can select one representative out of each equivalence class giving us a set ∑* of simple modules. Every simple module is isomorphic to exactly one element of ∑*. In all of what follows all the results could be rephrased in terms of ∑*. However, for the sake of simplicity and directness, we will use ∑ instead.

An ideal (such as a radical) could be described or characterized in the following three different ways: (i) by specifying what types of (one sided) ideals or what types of elements it necessarily must always contain; or (ii) as an intersection of certain types of (one sided) ideals, (iii) Necessary and sufficient conditions could be found in order for an arbitrary element of the ring to belong to the radical in question.