25 research outputs found
Relative injective modules, superstability and noetherian categories
We study classes of modules closed under direct sums,
-submodules and -epimorphic images where
is either the class of embeddings, -embeddings or pure
embeddings.
We show that the -injective modules of theses classes satisfy a
Baer-like criterion. In particular, injective modules, -injective modules,
pure injective modules, flat cotorsion modules and -torsion pure
injective modules satisfy this criterion. The argument presented is a model
theoretic one. We use in an essential way stable independence relations which
generalize Shelah's non-forking to abstract elementary classes.
We show that the classical model theoretic notion of superstability is
equivalent to the algebraic notion of a noetherian category for these classes.
We use this equivalence to characterize noetherian rings, pure semisimple
rings, perfect rings and finite products of finite rings and artinian valuation
rings via superstability.Comment: 25 page
Connected components of definable groups and o-minimality I
We give examples of groups G such that G^00 is different from G^000. We also
prove that for groups G definable in an o-minimal structure, G has a "bounded
orbit" iff G is definably amenable. These results answer questions of
Gismatullin, Newelski, Petrykovski. The examples also give new non G-compact
first order theories.Comment: 26 pages. This paper corrects the paper "Groups definable in
o-minimal structures: structure theorem, G^000, definable amenability, and
bounded orbits" by the first author which was posted in December
(1012.4540v1) and later withdraw