Exact embedding functors for module categories and submodule lattice quasivarieties
type: Item
instance of: scholarly article
title: Exact embedding functors for module categories and submodule lattice quasivarieties
zbMATH Open document ID: 0934.16007
author: George Hutchinson
published in: Journal of Algebra
publication date: 2000-04-09T00:00:00Z
review text: For rings with unit \(R\) and \(S\) the relation \(R\precsim S\) means that there exists an exact embedding functor \(F\colon R\text{-Mod}\to S\text{-Mod}\). By \({\mathcal L}(R)\) is denoted the quasi-variety of lattices generated by the family of all submodule lattices \(\text{Su}(_RM)\) for \(M\in R\text{-Mod}\) (a lattice \(L\) is in \({\mathcal L}(R)\) iff it is isomorphic to a sublattice of some \(\text{Su}(_RM)\)). It is known that \(R\precsim S\) iff \({\mathcal L}(R)\subseteq{\mathcal L}(S)\). In this article the lattice structure for submodule lattice quasivarieties is investigated. Let \(\mathcal W\) be the set of all quasivarieties of lattices. For a class of rings \({\mathcal R}'\) by \({\mathcal W}({\mathcal R}')\) is denoted the subset of \(\mathcal W\) consisting of all quasivarieties equal to \({\mathcal L}(R)\) for some \(R\in{\mathcal R}'\). For the class \({\mathcal R}_c\) of all commutative rings it is proved that \({\mathcal W}({\mathcal R}_c)\) is a complete lattice where \({\mathcal L}(R\times S)\) and \({\mathcal L}(R\otimes S)\) are the least upper bound (lub) and the greatest lower bound (glb), respectively, of \({\mathcal L}(R)\) and \({\mathcal L}(S)\) in \({\mathcal W}({\mathcal R}_c)\), and glb of an infinite family \(\{{\mathcal L}(R_j)\}_{j\in J}\) in \({\mathcal W}({\mathcal R}_c)\) can be formed by finite tensor products of rings in \(\{R_j\}_{j\in J}\). Further the rings \(R\) of characteristic zero are considered and for every prime \(p\) the associated ring \(R_p\) is constructed such that \({\mathcal L}(R_p)\subseteq{\mathcal L}(R)\) and \({\mathcal L}(R)\) is the join in \({\mathcal W}\) of \({\mathcal L}(\mathbb{Q})\) and \({\mathcal L}(R_p)\) for all primes \(p\). The relation \(R\precsim S\) is equivalent to \({\mathcal L}(R_p)\subseteq{\mathcal L}(S_p)\) for all primes \(p\) and \(\text{char}(R)\) divides \(\text{char}(S)\) or \(\text{char}(S)=0\).
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 08C15
zbMATH DE Number: 1346328
zbMATH Keywords: embedding functors
zbMATH Keywords: quasi-varieties of lattices
zbMATH Keywords: submodule lattices
reviewed by: A. I. Kashu
MaRDI profile type: Publication
full work available at URL: jabr.1999.7865
OpenAlex ID: W2075038810
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DOI: 10.1006/JABR.1999.7865
description: scientific article; zbMATH DE number 1346328