In abelian group theory, an abelian group is said to be cotorsion if every extension of it by a torsion-free group splits. If the group is , this says that for all torsion-free groups . It suffices to check the condition for the group of rational numbers. More generally, a module M over a ring R is said to be a cotorsion module if Ext1(F,M)=0 for all flat modules F. This is equivalent to the definition for abelian groups (considered as modules over the ring Z of integers) because over Z flat modules are the same as torsion-free modules. Some properties of cotorsion groups:
Attributes | Values |
---|
rdf:type
| |
rdfs:label
| |
rdfs:comment
| - In abelian group theory, an abelian group is said to be cotorsion if every extension of it by a torsion-free group splits. If the group is , this says that for all torsion-free groups . It suffices to check the condition for the group of rational numbers. More generally, a module M over a ring R is said to be a cotorsion module if Ext1(F,M)=0 for all flat modules F. This is equivalent to the definition for abelian groups (considered as modules over the ring Z of integers) because over Z flat modules are the same as torsion-free modules. Some properties of cotorsion groups: (en)
|
dcterms:subject
| |
Wikipage page ID
| |
Wikipage revision ID
| |
Link from a Wikipage to another Wikipage
| |
sameAs
| |
dbp:wikiPageUsesTemplate
| |
first
| |
id
| |
last
| |
title
| |
has abstract
| - In abelian group theory, an abelian group is said to be cotorsion if every extension of it by a torsion-free group splits. If the group is , this says that for all torsion-free groups . It suffices to check the condition for the group of rational numbers. More generally, a module M over a ring R is said to be a cotorsion module if Ext1(F,M)=0 for all flat modules F. This is equivalent to the definition for abelian groups (considered as modules over the ring Z of integers) because over Z flat modules are the same as torsion-free modules. Some properties of cotorsion groups:
* Any quotient of a cotorsion group is cotorsion.
* A direct product of groups is cotorsion if and only if each factor is.
* Every divisible group or injective group is cotorsion.
* The Baer Fomin Theorem states that a torsion group is cotorsion if and only if it is a direct sum of a divisible group and a , that is, a group of bounded exponent.
* A torsion-free abelian group is cotorsion if and only if it is algebraically compact.
* Ulm subgroups of cotorsion groups are cotorsion and Ulm factors of cotorsion groups are algebraically compact. (en)
|
oldid
| |
prov:wasDerivedFrom
| |
page length (characters) of wiki page
| |
foaf:isPrimaryTopicOf
| |
is Link from a Wikipage to another Wikipage
of | |
is Wikipage redirect
of | |
is foaf:primaryTopic
of | |