Modules with few types
type: Item
instance of: scholarly article
title: Modules with few types
zbMATH Open document ID: 0768.03017
DOI: 10.1016/0021-8693(92)90022-E
author: Ivo Herzog
published in: Journal of Algebra
publication date: 1992-09-27T00:00:00Z
review text: Let \(R\) be a countable ring. A left \(R\)-module \(M\) is said to have few types if there are just countably many complete types in a finite number of variables consistent with \(M\) (in the language of left \(R\)-modules). It is proved in this paper that if \(M\) has few types, then the pure- injective hull of a type in finitely many variables is a direct sum of finitely many indecomposable modules. It follows that \(M\) has few types iff the direct sum of infinitely many copies of \(M\) has few types. If the ring \(R\) is commutative, then the annihilator ideal of any subset of \(M\) is shown to have a primary decomposition. If \(M\) has few types, a regular generic type and is not definably connected-by-finite, it is proved that \(M\) must be weakly minimal. If the ring \(R\) is right Noetherian, we settle Vaught's Conjecture for a weakly minimal left \(R\)-module \(M\) by showing that if \(M\) has few types, then it must be totally transcendental.
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 13L05
zbMATH DE Number: 63617
zbMATH Keywords: left \(R\)-modules
zbMATH Keywords: pure-injective hull of a type
zbMATH Keywords: direct sum
zbMATH Keywords: indecomposable modules
zbMATH Keywords: annihilator ideal
zbMATH Keywords: primary decomposition
zbMATH Keywords: regular generic type
zbMATH Keywords: weakly minimal
zbMATH Keywords: Vaught's Conjecture
zbMATH Keywords: totally transcendental
MaRDI profile type: Publication
cites work: The classification of small weakly minimal sets. III: Modules
cites work: Decomposition of totally transcendental modules
cites work: Four concepts from “geometrical” stability theory in modules
cites work: Model theory of modules
description: scientific article; zbMATH DE number 63617