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

DOI10.1016/J.EXMATH.2022.12.003arXiv2209.00337MaRDI QIDQ6409445

Amit Shah

Publication date: 1 September 2022

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

Mathematics Subject Classification ID

Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) (16D70) Noncommutative local and semilocal rings, perfect rings (16L30) Abelian categories, Grothendieck categories (18E10) Preadditive, additive categories (18E05) Idempotent elements (associative rings and algebras) (16U40)

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