Complete homomorphisms between module lattices
type: Item
instance of: scholarly article
zbMATH Open document ID: 1307.13016
publication date: 2014-09-01T00:00:00Z
full work available at URL: 2-V16-2014.pdf
Mathematics Subject Classification ID: 13C99
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 06B10
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 6336952
zbMATH Keywords: lattice of ideals
zbMATH Keywords: lattice of submodules
zbMATH Keywords: multiplication modules
zbMATH Keywords: complete lattice
zbMATH Keywords: complete homomorphism
author: Patrick F. Smith
MaRDI profile type: Publication
OpenAlex ID: W2183171487
DOI: 10.24330/IEJA.266224
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title: Complete homomorphisms between module lattices
published in: International Electronic Journal of Algebra (IEJA)
review text: Let \(R\) be a commutative ring with identity and let \(M\) be a unital \(R\) -module, throughout. A ring \(R\) with Jacobson radical \(J\) is called a semiperfect ring in case \(R/J\) is semiprime Artinian allowing idempotents to lift modulo \(J,\) page 303 of \textit{F. W. Anderson} and \textit{K. R. Fuller}'s [Rings and categories of modules. Berlin: Springer-Verlag (1974; Zbl 0301.16001)]. An \(R\)-module \(M\) is called faithful if \(\mathrm{ann}_{R}(M)=(0)\) and \(M\) is a multiplication module if for every \(R\)-submodule \(N\) of \(M\) there is an ideal \(I\) of \(R\) such that \(N=IM.\) Denote by \(\mathcal{L(}_{R}M)\) the lattice of \(R\)-submodules of \(M.\) Regarding \(R\) as an \(R\)-module we can consider the lattice \(\mathcal{L(} _{R}R)\) of ideals of \(R\). Usually \(\mathcal{L(}_{R}\mathcal{R})\) is denoted by \(\mathcal{L(R}).\) Recall that for \(A,B\) in \(\mathcal{L(}_{R}M),A\vee B=A+B \) and \(A\wedge B=A\cap B.\) A lattice \(\mathcal{L}\) is said to be complete if for every nonemplty subset \(S\) of \(\mathcal{L}\) the least upper bound \(\vee (S)\) and the greatest lower bound \(\wedge (S))\) exist and are in \(\mathcal{L} \). Indeed \(\mathcal{L(}_{R}M)\) and \(\mathcal{L(R})\) are complete lattices.NEWLINENEWLINEGiven two lattices \(\mathcal{L}\), \(\mathcal{L}^{\prime }\) a map \(\varphi : \mathcal{L}\rightarrow \mathcal{L}^{\prime }\) is called a homomorphism if for all \(a,b\in \mathcal{L}\) we have \(\varphi (a\vee b)=\varphi (a)\vee \varphi (b)\) and \(\varphi (a\wedge b)=\varphi (a)\wedge \varphi (b).\) If \( \mathcal{L}\) and \(\mathcal{L}^{\prime }\) are both complete a homomorphism \( \varphi :\mathcal{L}\rightarrow \mathcal{L}^{\prime }\) is is called a complete homomorphism if for every nonempty subset \(S\) of \(\mathcal{L}\) we have \(\varphi (\vee (S))=\vee (\varphi (S))=\vee \{\varphi (x):x\in S\}\) and \(\varphi (\wedge (S))=\wedge (\varphi (S))=\wedge \{\varphi (x):x\in S\}.\) The paper under review is essentially to do wit…
reviewed by: Muhammad Zafrullah
description: scientific article; zbMATH DE number 6336952