Uniqueness of uniform decompositions in abelian categories
type: Item
instance of: scholarly article
title: Uniqueness of uniform decompositions in abelian categories
zbMATH Open document ID: 1027.18007
DOI: 10.1016/S0022-4049(03)00064-1
author: Henning Krause
published in: Journal of Pure and Applied Algebra
publication date: 2003-11-17T00:00:00Z
review text: The author proves a Krull-Schmidt type theorem for finite direct sums of uniform objects in an abelian category \(\mathcal A\). If \(\mathcal A\) has injective envelopes, the decomposition theorem turns into the Krull-Schmidt theorem for injectives. Therefore, the proof is achieved by embedding \(\mathcal A\) into the Grothendieck category \(\text{Lex}(\mathcal A)\) of left exact functors from \({\mathcal A}^{\text{op}}\) to \({\mathcal A}b\). As a Grothendieck category, \(\text{Lex}(\mathcal A)\) has injective envelopes. The main point of the paper is the observation, that a recent unique decomposition theorem of Diracca and Facchini which replaces isomorphism between injective envelopes of objects \(A, B\) by the existence of monomorphisms \(A\hookrightarrow B\hookrightarrow A\), arises as a special case of the author's result.
reviewed by: Wolfgang Rump
Mathematics Subject Classification ID: 18E40
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 18F15
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 2005424
zbMATH Keywords: Krull-Schmidt type theorem
zbMATH Keywords: uniform objects in abelian category
zbMATH Keywords: Grothendieck category
zbMATH Keywords: unique decomposition theorem
MaRDI profile type: Publication
cites work: Uniqueness of monogeny classes for uniform objects in abelian categories
cites work: Des catégories abéliennes
full work available at URL: s0022-4049(03)00064-1
OpenAlex ID: W1995233763
description: scientific article; zbMATH DE number 2005424