City Research Online

Abstract

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 A for which there is a commutative unital ring k, such that each Hom-set in 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.

Publication Type:	Article
Additional Information:	This is an open access article distributed under the terms of the Creative Commons CC-BY license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Publisher Keywords:	Additive category, Hom-finite category, Idempotent, Krull-Remak-Schmidt decomposition, Krull-Schmidt category, Split idempotents
Subjects:	Q Science > QA Mathematics
Departments:	School of Science & Technology School of Science & Technology > Department of Mathematics
SWORD Depositor:	Symplectic Administrator

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