Rings over which all modules of a given type are almost categorical
type: Item
instance of: scholarly article
title: Rings over which all modules of a given type are almost categorical
zbMATH Open document ID: 0584.03025
DOI: 10.1007/BF01979704
author: E. M. Kremer
published in: Algebra and Logic
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/187261
review text: The author's characterization of almost categorical modules in the article reviewed above [see Zbl 0584.03024] is used for the study of rings with the condition of almost categoricity of all modules from one of the following classes: a) class of all R-modules, b) class of all injective R-modules, c) class of all projective R-modules (R is an associative ring with unity). Theorem 1. The following conditions are equivalent: 1) All left R-modules are almost categorical. 2) All right R-modules are almost categorical. 3) The ring R is Artinian and semisimple. Theorem 2. All injective left R-modules are almost categorical if and only if R is left Artinian. Theorem 3. All projective left R-modules are almost categorical if and only if R is a left perfect and right coherent ring. As corollaries, rings with the condition of categoricity of all modules from one of the above classes are characterized. It is also shown that if K is one of the classes b) or c), then the following conditions are equivalent: 1) K is categorical, 2) K is axiomatizable and complete, 3) K is axiomatizable and model complete.
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 08C10
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 03C35
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 3935019
zbMATH Keywords: almost categoricity
zbMATH Keywords: injective left R-modules
zbMATH Keywords: projective left R-modules
zbMATH Keywords: left perfect and right coherent ring
MaRDI profile type: Publication
cites work: Almost categorical theories
cites work: A non-two-cardinal set of stable types
cites work: Q3043320
cites work: Q5682108
cites work: Q4141349
cites work: Locally graded groups, factorizable by subgroups of finite rank
cites work: Decomposition of totally transcendental modules
cites work: Discussion of some objections to thermomechanical orthogonality
cites work: Elimination of quantifiers for modules
cites work: Model-completions and modules
cites work: Definability problems for modules and rings
OpenAlex ID: W2068485483
description: scientific article; zbMATH DE number 3935019