The construction of mixed modules from a theorem of linear algebra
type: Item
instance of: scholarly article
title: The construction of mixed modules from a theorem of linear algebra
zbMATH Open document ID: 0629.20026
DOI: 10.1016/0021-8693(87)90044-5
author: Rüdiger Göbel
author: Warren May
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let R be a commutative ring with \(0\neq 1\in R\) and \(1\in S\subseteq R\) a countable, multiplicatively closed subset of R without zero divisors. We say that an R-module M is S-torsion if \(t_ SM=M\) and M is S-torsion- free if \(t_ SM=0\), where \(t_ SM=\cup_{s\in S}M[s]\), \(M[s]=\{m\in M|\) \(ms=0\}\). We call M S-reduced, if \(0=\cap_{s\in S}Ms\). Let us denote the obvious localization of R by \(S^{-1}R\). Let (1) A be an S-torsion-free R-algebra, where S-torsion-free refers to the R-module structure of A. We choose a fixed set of n multiplicatively closed subsets \(S_ i\) of S \((i<n)\) \(with\) (2)As\(+At=A\) for all \(s\in S_ i\), \(t\in S_ j\), \(i\neq j<n.\) Further, suppose that (3) A is \(S_ i\)-reduced for all \(i<n.\) Finally (4) Each \(s\in S\) is a product of elements in \(\cup_{i<n}S_ i.\) If k is a (finite or infinite) cardinal, then the authors construct in their main theorems (2.2, 2.3, \(2.3^*)\) families \(F=F(k)\) of mixed R- modules G with additional properties: (5) (a) There exists a direct sum \(T=\oplus_{i<n}T_ i\) of n direct sums \(T_ i\) of \(S_ i\)-bounded (cyclic A-)modules such that each \(G\in F\) has \(t_ sG=T.\) (b) G is S-reduced and \(G/T=\oplus_ kS^{-1}A\) is a free \(S^{-1}A\)- module of rank k. (c) End G\(=A\oplus Hom_ R(G,T).\) (d) If G,H\(\in F\), then \(Hom(G,T)=Hom(H,T)\) and End \(G\cong End H\) as R- algebras. (e) If \(G\neq H\in F\), then \(Hom(G,H)=Hom(G,T).\) Theorem 2.2. Let k be an infinite cardinal. If A is an R-algebra as above, and R has \(n\geq 5\) sets \(S_ i\) (respectively, \(A=R\) and \(n\geq 4)\), then there exists a family \(F=F(k)\) of size \(2^ k\) satisfying (5). A set \(X\subseteq A\) is a quasi-generator of the algebra A if \(Hom_ R(A/A(X),A)=0\) for the subalgebra A(X) of A generated by X. Let q(A) denote the minimal cardinal \(| X|\) of all possible quasi- generators X of A. Theorem 2.3. Suppose k is a finite cardinal and A an R-algebra as above. a) Let \(k=2\), \(A=R\), and \(n\geq 3\). b) Let \(…
reviewed by: A. M. Sebel'din
Mathematics Subject Classification ID: 20K21
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13C13
Mathematics Subject Classification ID: 20K25
zbMATH DE Number: 4023569
zbMATH Keywords: S-torsion-free R-algebra
zbMATH Keywords: multiplicatively closed subsets
zbMATH Keywords: mixed R- modules
zbMATH Keywords: direct sums
zbMATH Keywords: quasi-generator
zbMATH Keywords: Dedekind domain
zbMATH Keywords: torsion submodule
zbMATH Keywords: endomorphism algebras
zbMATH Keywords: direct sum of bounded R-modules
zbMATH Keywords: Kaplansky's test problems
zbMATH Keywords: mixed modules
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90044-5
OpenAlex ID: W2062606910
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description: scientific article; zbMATH DE number 4023569