In category theory, a branch of mathematics, a KrullâÂÂSchmidt category is a generalization of categories in which the KrullâÂÂSchmidt theorem holds. They arise, for example, in the study of finite-dimensional modules over an algebra.
Let C be an additive category, or more generally an additive -linear category for a commutative ring . We call C a KrullâÂÂSchmidt category provided that every object decomposes into a finite direct sum of objects having local endomorphism rings. Equivalently, C has split idempotents and the endomorphism ring of every object is semiperfect.
One has the analogue of the KrullâÂÂSchmidt theorem in KrullâÂÂSchmidt categories:
An object is called indecomposable if it is not isomorphic to a direct sum of two nonzero objects. In a KrullâÂÂSchmidt category we have that
One can define the AuslanderâÂÂReiten quiver of a KrullâÂÂSchmidt category.
The category of finitely-generated projective modules over the integers has split idempotents, and every module is isomorphic to a finite direct sum of copies of the regular module, the number being given by the rank. Thus the category has unique decomposition into indecomposables, but is not Krull-Schmidt since the regular module does not have a local endomorphism ring.