Abstract
If A is right Artinian, then the finitely generated right A-modules can be constructed in an orderly way from the indecomposable modules, and the construction is unique by the Krull—Schmidt Theorem. The next step toward understanding A-modules is therefore in the direction of indecomposable modules, and this topic is currently the center of vigorous activity in ring theory. The aim of this chapter and the next chapter is to introduce the reader to the flavor of two hnes that are being pursued by research mathematicians who are now working on the theory of modules.
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