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An additive category in which each object has a Krull-Remak-Schmidtdecomposition -- that is, a finite direct sum decomposition consisting ofobjects with local endomorphism rings -- is known as a Krull-Schmidt category.A Hom-finite category is an additive category $\mathcal{A}$ for which there isa commutative unital ring $k$, such that each Hom-set in $\mathcal{A}$ is afinite length $k$-module. The aim of this note is to provide a proof that aHom-finite category is Krull-Schmidt, if and only if it has split idempotents,if and only if each indecomposable object has a local endomorphism ring.

Krull-Remak-Schmidt decompositions in Hom-finite additive categories