Four submodules suffice for realizing algebras over commutative rings
type: Item
instance of: scholarly article
title: Four submodules suffice for realizing algebras over commutative rings
zbMATH Open document ID: 0716.16015
DOI: 10.1016/0022-4049(90)90098-3
author: Rüdiger Göbel
author: Warren May
published in: Journal of Pure and Applied Algebra
publication date: 1990-01-01T00:00:00Z
review text: Let R be a commutative ring with identity and for \(n\geq 0\), \(R_ n mod\) denote the category of R-modules M with n distinguished submodules. A typical object is denoted by \(M=(M,M^ k;k<n)\). The main result of this paper is the following: Let \(\lambda\) be an infinite cardinal and let A be an R-algebra generated by \(\leq \lambda\) elements. Let M be equal to \(\oplus_{\lambda}A\). Then there exists an \(R_ 4\) module \(M=(M,M^ 0,M^ 1,M^ 2,M^ 3)\), with \(A=End M\), where End M consists of all R linear endomorphisms of M which leave \(M^ k\) stable, \(0\leq k\leq 3\). As a Corollary it follows that if A is a K-algebra, of dimension \(\leq \lambda\), an infinite cardinal, where K is a field then there exists a \(K_ 4\)-module M with M of dimension \(\lambda\) with \(A\cong End M\).
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 4179583
zbMATH Keywords: realization
zbMATH Keywords: infinite rank four submodules problem
zbMATH Keywords: free A-module
zbMATH Keywords: fully rigid family
zbMATH Keywords: category of R-modules
zbMATH Keywords: distinguished submodules
zbMATH Keywords: \(R_ 4\) module
zbMATH Keywords: linear endomorphisms
zbMATH Keywords: \(K_ 4\)-module
MaRDI profile type: Publication
full work available at URL: 0022-4049(90)90098-3
OpenAlex ID: W2007763357
cites work: Q4734019
cites work: On the elementary theory of quadruples of vector spaces
cites work: Endomorphism algebras of vector spaces with distinguished sets of subspaces
cites work: Q4777469
cites work: Endomorphism Rings of Vector Spaces and Torsion Free Abelian Groups
cites work: Pathological Modules over Tame Rings
cites work: Q3798883
cites work: Endomorphism algebras of large modules with distinguished submodules
cites work: Prescribing Endomorphism Algebras, a Unified Treatment
cites work: Q3819239
cites work: Note on extensions of Abelian groups by primary groups
cites work: Large indecomposable modules in torsion theories
cites work: Q4046184
cites work: Vector spaces with five distinguished subspaces
cites work: The construction of mixed modules from a theorem of linear algebra
cites work: Q3834196
cites work: Very decomposable Abelian groups
cites work: Q3866236
cites work: Tame algebras and integral quadratic forms
cites work: Infinite Abelian groups, Whitehead problem and some constructions
Recommended article: Q4505812
Recommended article: Q5241306
Recommended article: Q2990529
Recommended article: Q4406612
Recommended article: On the commutative rings with at most two proper subrings
Recommended article: Q4788506
Recommended article: Q3781894
Recommended article: Four-dimensional real division algebras with few derivations
Recommended article: Classification of the four-dimensional power-commutative real division algebras
Recommended article: Four-dimensional real algebras satisfying left or right ¢ -associativity
description: scientific article; zbMATH DE number 4179583