Abstract
This chapter is the beginning of an examination of algebras that are more general than the semisimple algebras. Our strategy is to generalize the process that led to the Wedderburn Structure Theorem. The appropriate substitutes for the basic building blocks of that theory—simple modules—are indecomposable modules. These modules are introduced in this chapter. The analogue of Schur’s lemma provides a characterization of indecomposable modules in terms of their endomorphism algebras. The main result of the chapter is the Krull-Schmidt Theorem. It leads to the conclusion that finitely generated modules over an Artinian algebra decompose uniquely into direct sums of indecomposable modules. In short, the results of Chapter 2 have close analogues in the theory of modules over Artinian algebras.
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