About: Cotorsion group     Goto   Sponge   NotDistinct   Permalink

An Entity of Type : yago:WikicatPropertiesOfGroups, within Data Space : dbpedia.org associated with source document(s)
QRcode icon
http://dbpedia.org/describe/?url=http%3A%2F%2Fdbpedia.org%2Fresource%2FCotorsion_group

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:

AttributesValues
rdf:type
rdfs:label
  • Cotorsion group (en)
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
  • L. (en)
id
  • Cotorsion_group (en)
last
  • Fuchs (en)
title
  • Cotorsion group (en)
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
Faceted Search & Find service v1.17_git139 as of Feb 29 2024


Alternative Linked Data Documents: ODE     Content Formats:   [cxml] [csv]     RDF   [text] [turtle] [ld+json] [rdf+json] [rdf+xml]     ODATA   [atom+xml] [odata+json]     Microdata   [microdata+json] [html]    About   
This material is Open Knowledge   W3C Semantic Web Technology [RDF Data] Valid XHTML + RDFa
OpenLink Virtuoso version 08.03.3330 as of Mar 19 2024, on Linux (x86_64-generic-linux-glibc212), Single-Server Edition (61 GB total memory, 55 GB memory in use)
Data on this page belongs to its respective rights holders.
Virtuoso Faceted Browser Copyright © 2009-2024 OpenLink Software