Premium
Indecomposable flat cotorsion modules
Author(s) -
Guil Asensio Pedro A.,
Herzog Ivo
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm076
Subject(s) - functor , indecomposable module , mathematics , morphism , pure mathematics , simple (philosophy) , flat module , ring (chemistry) , simple module , abelian category , monomorphism , abelian group , finitely generated abelian group , projective module , injective function , chemistry , philosophy , organic chemistry , epistemology
An additive functor from the category of flat right R ‐modules to the category of abelian groups is continuous if it is isomorphic to a functor of the form− ⊗ R M , where M is a left R ‐module. It is shown that for any simple subfunctor A of−⊗ M there is a unique indecomposable flat cotorsion module U R for which A ( U )≠0. It is also proved that every subfunctor of a continuous functor contains a simple subfunctor. This implies that every flat right R ‐module may be purely embedded into a product of indecomposable flat cotorsion modules. If CE( R ) is the cotorsion envelope of R R and S = End ; R CE( R ), then a local ring monomorphism is constructed from R / J ( R ) to S / J ( S ). This local morphism of rings is used to associate a semiperfect ring to any semilocal ring. It also proved that if R is a semilocal ring and M a simple left R ‐module, then the functor− ⊗ R M on the category of flat right R ‐modules is uniform, and therefore contains a unique simple subfunctor.