Introduction

This chapter contains more major results of module theory than any previous chapter. It discusses chain condition in modules and direct sum decompositions of modules. The two are closely connected. Chain conditions in modules and rings have significant influence on the structure of a module or ring. For example, the D.C.C. together with semiprimitivity leads to the Wedderburn Theorems. And a ring R is right Artin semiprimitive if and only if every module is a direct sum of simple modules. In another direction, we saw that a ring R was right Noetherian if and only if every direct sum of injective modules remains injective. Note that in both examples above direct sum decompositions of modules appear together with chain conditions.

So far we understand semiprimitive right Artinian rings R completely through the Wedderburn Theorems. At this point quite naturally, two questions arise. What happens if we drop the semiprimitivity restriction? And what can we say about the structure of a right Noetherian ring, with some appropriate auxiliary restrictions in addition to the A.C.C.? Both questions are answered by the Hopkins–Levitzki Theorem (11–3.8). Hopkins proved that a right Artinian ring is right Noetherian. However, the latter fact alone does not give us a picture of what an arbitrary right Artinian ring looks like. Here a theorem of Levitzki supplies the missing pieces in this jigsaw puzzle. Levitzki Theorem (11–3.6) states that in a right Noetherian ring, every nil one-sided ideal is nilpotent.