The Krull-Schmidt theorem

Abstract

Publisher Summary This chapter provides an overview of the basic concepts of the Krull-Schmidt theorem. The two proofs of the Krull-Schmidt-Azumaya theorem, mention refinements of decompositions, and the exchange property and study endomorphism rings of modules of finite length. The validity of the Krull-Schmidt theorem, for finite direct sum decompositions, is a property of the endomorphism ring. The “classical” Krull-Schmidt theorem asserts that any two direct sum decompositions of a module of finite length into indecomposable summands are isomorphic. The Krull-Schmidt theorem holds for the class of modules of finite length. The basic properties of semi-perfect rings and semi-local rings, because endomorphism rings of modules of finite length are semi-perfect rings, and semi-perfect rings are semi-local. The validity of the Krull-Schmidt theorem for modules over commutative rings is surveyed. A second class of almost Krull- Schmidt modules, the modules whose endomorphism ring is semi-local is reviewed. The biuniform modules in general and uniserial modules in particular are focused. The isomorphism classes of finitely generated projective modules over a semi-local ring form a commutative monoid that is isomorphic to a full submonoid of the additive monoid N”. Some results about direct sums of modules whose endomorphism rings are homogeneous semi-local are presented.