Krull dimension of bimodules.
type: Item
instance of: scholarly article
title: Krull dimension of bimodules.
zbMATH Open document ID: 1117.16011
published in: Journal of Algebra
publication date: 2007-04-18T00:00:00Z
review text: Let \(R\) and \(S\) be unital rings with identity, and let \(M\) be a left \(S\)-, right \(R\)-bimodule. One says that the bimodule \(_SM_R\) is `Krull symmetric' if the modules \(_SM\) and \(M_R\) both have Krull dimension, denoted by \(\text{k}(_SM)\) and \(\text{k}(M_R )\), respectively, and \(\text{k}(_S(M/N))=\text{k}((M/N)_R)\) for all sub-bimodules \(N\) of \(_SM_R\). The authors prove that if \(_SM_R\) is a Krull symmetric bimodule such that the module \(_SM\) is finitely generated then \(\text{k}(M_R)=\text{k}((M/K)_R)\) for some prime submodule \(K\) of \(M_R\) such that \((M/K)_R\) is k-critical. The authors also prove the following bimodule analogue of a result from 1973 of Lambek and Michler: if \(_SM_R\) is a Noetherian bimodule (this means that \(M\) is Noetherian both as a left \(S\)-module and as a right \(R\)-module), then the right \(R\)-module \(M\) is Artinian if and only if every irreducible prime submodule of \(M_R\) is maximal.
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16D20
zbMATH DE Number: 5144399
zbMATH Keywords: Artinian rings
zbMATH Keywords: Artinian modules
zbMATH Keywords: Noetherian rings
zbMATH Keywords: Noetherian modules
zbMATH Keywords: Krull dimension
zbMATH Keywords: bimodules
author: Patrick F. Smith
author: Alex R. Woodward
reviewed by: Toma Albu
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2007.01.013
OpenAlex ID: W2018645062
cites work: Q3154382
cites work: Q4055809
cites work: Krull dimension
cites work: The torsion theory at a prime ideal of a right Noetherian ring
cites work: Prime submodules of Noetherian modules
cites work: Q3786586
cites work: Q4673772
DOI: 10.1016/J.JALGEBRA.2007.01.013
description: scientific article; zbMATH DE number 5144399