论文标题
krull-remak-schmidt分解hom-finite添加剂类别
Krull-Remak-Schmidt decompositions in Hom-finite additive categories
论文作者
论文摘要
每个对象都有一个krull-remak-schmidt分解的加性类别 - 即，由具有局部内态环的对象组成的有限直接总和分解 - 被称为krull-schmidt类别。 HOM-FINITE类别是一个添加类别$ \ MATHCAL {a} $，其中有一个交换性的Unital Ring $ K $，因此每个HOM set in $ \ Mathcal {a} $都是有限的长度$ K $ -MODULE。本说明的目的是提供一个证明，当且仅当它具有拆分iDempotents时，当且仅当每个不可分解的对象都具有局部内态态度时，且仅当它具有分裂的iDempotents时，就提供了一个证据。
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.
