Some natural additive decompositions of elements in bimodules
type: Item
instance of: scholarly article
title: Some natural additive decompositions of elements in bimodules
zbMATH Open document ID: 1477.16007
author: Alberto Facchini
author: Francesco Pagliuca
author: Adel Alahmadi
published in: Journal of Pure and Applied Algebra
publication date: 2021-09-02T00:00:00Z
review text: Let \(R\) and \(S\) be unital rings and \(_RM_S\) a unital \(R-S\)-bimodule. The authors define a category \(\mathcal C\) whose objects are the elements of \(M\) and whose morphisms are elements of \(R\times S\), defined for all \(x,\, y\in M\textrm{ by Hom}_\mathcal C(x,y)=\{(r,s)\colon rx=sy\}\), so that for all \(x\in M,\ \textrm{End}_\mathcal C(x)\) is a subring of \(R\times S\). Factoring by the ideal \(\mathcal I(x,y)=\{(r,s)\colon rx=0=ys\}\) yields a category \(\mathcal D\) which is the primary focus of study in this paper. Using a mix of module and category theoretic techniques, the authors show that \(\mathcal D\) is a pre-additive category in which idempotents split. It follows that for all \(x\in {}_RM_S\) there is a 1-1 correspondence between idempotent elements in End\(_{\mathcal D}(x)\) and the set of all pairs \((x_1, x_2)\) such that \(x=x_1\oplus x_2\). Several examples include \(R=M=R \) and \(M\) Artinian with endomorphism ring \(R\), in which case decompositions of 1 into pairwise orthogonal primitive idempotents correspond to indecomposable decompositions of \(R\). The second half of the paper relates these results to various functors between categories whose objects have semi-local endomorphism rings.
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16D20
Mathematics Subject Classification ID: 16L30
zbMATH DE Number: 7389903
zbMATH Keywords: bimodule
zbMATH Keywords: additive decomposition
zbMATH Keywords: semilocal ring
reviewed by: Phillip Schultz
MaRDI profile type: Publication
full work available at URL: j.jpaa.2021.106806
OpenAlex ID: W3176566343
cites work: Direct products of modules whose endomorphism rings have at most two maximal ideals
cites work: Equivalence of diagonal matrices over local rings.
cites work: Q4023148
cites work: Factorizations of elements in local rings and semilocal rings of finite type
cites work: Homomorphisms with semilocal endomorphism rings between modules
cites work: On semilocal rings
cites work: Q4789144
cites work: Q3282017
cites work: Krull-Schmidt fails for serial modules
cites work: Direct sum decompositions of modules, semilocal endomorphism rings, and Krull monoids
cites work: Direct-sum decompositions of modules with semilocal endomorphism rings
cites work: Semilocal Categories and Modules with Semilocal Endomorphism Rings
cites work: Krull-Schmidt Fails for Artinian Modules
cites work: Q4950132
cites work: Q4215784
cites work: Intrinsic Characterizations of Some Additive Functors
DOI: 10.1016/J.JPAA.2021.106806
description: scientific article; zbMATH DE number 7389903