An additive category in which each object has a Krull-Remak-Schmidt decomposition—that is, a finite direct sum decomposition consisting of objects with local endomorphism rings—is known as a Krull-Schmidt category. A $Hom$-finite category is an additive category $\mathcal{A}$ for which there is a commutative unital ring $k$, such that each $Hom$-set in $\mathcal{A}$ is a finite length $k$-module. The aim of this note is to provide a proof that a $Hom$-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.

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Hom 有限附加类别中的 Krull-Remak-Schmidt 分解

每个对象都具有 Krull-Remak-Schmidt 分解的加性类别（即，由具有局部自同态环的对象组成的有限直和分解）称为 Krull-Schmidt 类别。A$坎$-有限范畴是一个附加范畴$\mathcal{A}$其中有一个交换单位环$k$, 这样每个$坎$-设置$\mathcal{A}$是有限长度$k$-模块。本说明的目的是提供一个证明$坎$- 有限范畴是 Krull-Schmidt，当且仅当它具有分裂幂等性，当且仅当每个不可分解的对象都具有局部自同态环。