Finitely generated, non-Artinian monolithic modules.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1258.16002
author: Ian M. Musson
published in: New Trends in Noncommutative Algebra
publication date: 2012-07-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/1104.1366
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16S15
zbMATH DE Number: 6052239
zbMATH Keywords: monolithic modules
zbMATH Keywords: injective hulls
zbMATH Keywords: simple modules
zbMATH Keywords: locally Artinian modules
zbMATH Keywords: down-up algebras
MaRDI profile type: Publication
OpenAlex ID: W1536895759
arXiv ID: 1104.1366
DOI: 10.1090/CONM/562/11138
title: Finitely generated, non-Artinian monolithic modules.
review text: The paper under review provides a brief survey of the study of Noetherian rings over which the injective hull of every simple module is locally Artinian. Recall that a module is called locally Artinian if each of its finitely generated submodules is Artinian. For any ring \(R\), the condition that the injective hulls of simple right \(R\)-modules are locally Artinian is obviously equivalent to the condition that all finitely generated essential extensions of simple right \(R\)-modules are Artinian. By Matlis' theorem it follows clearly that every commutative Noetherian ring satisfies the property that injective hulls of simple right \(R\)-modules are locally Artinian. A. V. Jategaonkar and J. E. Roseblade showed that if \(G\) is a polycyclic-by-finite group then injective hulls of simple modules over the group ring \(R[G]\) are locally Artinian whenever \(R\) is a ring of characteristic zero, or \(R\) is a field that is algebraic over a finite field. Dahlberg showed that injective hulls of simple modules over \(U(\mathfrak{sl}_2)\) are locally Artinian. Jategaonkar showed that for a fully bounded Noetherian ring \(R\), the injective hull of every simple \(R\)-module is locally Artinian, and he used this fact to show that Jacobson's conjecture holds for a fully bounded Noetherian ring. A module \(M\) is called `monolithic' if the intersection of all nonzero submodules of \(M\) is nonzero (in other words, \(M\) has a unique minimal submodule). Note that for a Noetherian ring \(R\), the property that the injective hull of every simple \(R\)-module is locally Artinian is equivalent to the condition that every finitely generated monolithic \(R\)-module is Artinian. The author provides a general construction for finitely generated, non-Artinian, monolithic modules.NEWLINENEWLINE Georgia Benkart and Tom Roby introduced a unital associative algebra \(A=A(\alpha,\beta,\gamma)\) over a field \(K\) with generators \(u\), \(d\) and defining relations \(d^2u=\alpha dud+\beta ud^…
reviewed by: Ashish K. Srivastava
description: scientific article; zbMATH DE number 6052239