Injective Modules, Projective Modules, and Their Decompositions

Abstract

In this concluding chapter we return to the study of decompositions of modules—specifically of injective and projective modules. First we examine characterizations of noetherian rings in terms of the structure of injective modules. Then, after considering the decomposition theory of direct sums of countably generated modules, we proceed to the study of semiperfect and perfect rings (those over which all finitely generated modules and, respectively, all modules have projective covers). In the final section we show that the structure of the endomorphism ring of a finitely generated module determines whether direct sums of copies of that module have decompositions that complement direct summands.