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 R-linear category for a commutative ring R. We call C a Krull–Schmidt 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 Krull–Schmidt theorem in Krull–Schmidt categories: An object is called indecomposable if it is not isomorphic to a direct sum of two nonzero objects.
Thus the category has unique decomposition into indecomposables, but is not Krull-Schmidt since the regular module does not have a local endomorphism ring.