Lattice preradicals with applications to Grothendieck categories and torsion theories.
type: Item
instance of: scholarly article
title: Lattice preradicals with applications to Grothendieck categories and torsion theories.
zbMATH Open document ID: 1327.06009
author: Mihai Iosif
author: Toma Albu
published in: Journal of Algebra
publication date: 2015-10-13T00:00:00Z
review text: The authors define a lattice preradical. This is based on the notion of a linear morphism of lattices introduced earlier by them. As the name suggests, such morphisms for lattices capture the fundamental properties of a linear mapping between modules. At first a preradical is defined on the category \(\mathcal{LM}\) of linear modular lattices. The main result here is that this notion is compatible with direct joins in the sense that a preradical commutes with arbitrary direct joins. The definition of a preradical is then slightly generalized to hereditary full subcategories of \(\mathcal{LM}\). This generalization is used to give lattice theoretic counterparts of the well-known module preradicals trace and reject. The other well-known radicals like the socle and the Jacobson radical are also introduced in this general setting and the relationships between them are explored. Finally it is shown that this approach to preradicals for lattices lends itself to a smooth and easy interpretation in Grothendieck categories as well as module categories with a hereditary torsion theory.
reviewed by: Stefan Veldsman
Mathematics Subject Classification ID: 06C05
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 06B35
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 18E15
zbMATH DE Number: 6493632
zbMATH Keywords: modular lattices
zbMATH Keywords: complete lattices
zbMATH Keywords: compactly generated lattices
zbMATH Keywords: upper continuous lattices
zbMATH Keywords: linear modular lattices
zbMATH Keywords: preradicals
zbMATH Keywords: atoms
zbMATH Keywords: coatoms
zbMATH Keywords: socles
zbMATH Keywords: radicals
zbMATH Keywords: Jacobson radical
zbMATH Keywords: trace
zbMATH Keywords: reject
Wikidata QID: Q116296819
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2015.07.030
OpenAlex ID: W1917447294
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DOI: 10.1016/J.JALGEBRA.2015.07.030
description: scientific article; zbMATH DE number 6493632