When are homogeneous functions linear? A lattice point of view.
type: Item
instance of: scholarly article
title: When are homogeneous functions linear? A lattice point of view.
zbMATH Open document ID: 1084.16025
DOI: 10.1007/BF03323017
author: Marcel Wild
author: C. J. Maxson
published in: Results in Mathematics
publication date: 2005-06-09T00:00:00Z
review text: Throughout, \(R\) is a ring with identity and all \(R\)-modules are unital left modules. If \(V\) and \(W\) are \(R\)-modules, a function \(f\colon V\to W\) is said to be \(R\)-homogeneous if \(f(rv)=rf(v)\) for all \(r\in R\) and \(v\in V\). Following terminology introduced by \textit{U. Albrecht} and the reviewer [Proc. Am. Math. Soc. 123, No. 8, 2381-2389 (1995; Zbl 0839.16024)], \(V\) is a ray if for every \(R\)-module \(W\) and every \(R\)-homogeneous \(f\colon V\to W\), the map \(f\) is additive, i.e. \(f\) is an \(R\)-module homomorphism. The authors study relationships between the submodule lattice of \(V\) and the condition that \(V\) be a ray. Their results are applied to further determine the structure of those rings \(R\) for which every \(R\)-module is a ray. A characterization of all semisimple modules which are rays is obtained.
reviewed by: Jutta Hausen
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16D10
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 2174566
zbMATH Keywords: submodule lattices
zbMATH Keywords: homogeneous functions
zbMATH Keywords: rays
zbMATH Keywords: automorphisms
zbMATH Keywords: semisimple modules
MaRDI profile type: Publication
cites work: Structural matrix algebras and their lattices of invariant subspaces
cites work: Non-Singular Modules and R-Homogeneous Maps
cites work: On Rings for which Homogeneous Maps are Linear
cites work: Q4693994
cites work: Forcing linearity numbers for projective modules
cites work: Forcing linearity numbers
cites work: Forcing linearity numbers for modules over simple domains.
cites work: Forcing linearity numbers for finitely generated modules
cites work: Q3356479
cites work: Q3786586
cites work: Q4040049
cites work: Finitely Generated Artinian and Distributive Modules are Cyclic
cites work: The fundamental theorem of projective geometry for an arbitrary length two module.
description: scientific article; zbMATH DE number 2174566