A note on clean abelian groups
When the intrinsic algebraic entropy is not really intrinsic
The intrinsic algebraic entropy ent(ɸ) of an endomorphism ɸ of an Abelian group G can be computed using fully inert subgroups of ɸ-invariant sections of G, instead of the whole family of ɸ-inert subgroups. For a class of groups containing the groups of finite rank, aswell as those groupswhich are trajectories of finitely generated subgroups, it is proved that only fully inert subgroups of the group itself are needed to comput ent(ɸ). Examples show how the situation may be quite different outside...
Algebraic entropies, Hopficity and co-Hopficity of direct sums of Abelian Groups
Necessary and sufficient conditions to ensure that the direct sum of two Abelian groups with zero entropy is again of zero entropy are still unknown; interestingly the same problem is also unresolved for direct sums of Hopfian and co-Hopfian groups.We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the groups. There are clear similarities between the various cases but there is not a simple duality involved.
Sir William Rowan Hamilton
Cotorsion-free algebras as endomorphism algebras in - the discrete and topological cases
The discrete algebras over a commutative ring which can be realized as the full endomorphism algebra of a torsion-free -module have been investigated by Dugas and Göbel under the additional set-theoretic axiom of constructibility, . Many interesting results have been obtained for cotorsion-free algebras but the proofs involve rather elaborate calculations in linear algebra. Here these results are rederived in a more natural topological setting and substantial generalizations to topological...
Some generalizations of torsion-free Crawley groups
In this paper we investigate two new classes of torsion-free Abelian groups which arise in a natural way from the notion of a torsion-free Crawley group. A group is said to be an Erdős group if for any pair of isomorphic pure subgroups with , there is an automorphism of mapping onto ; it is said to be a weak Crawley group if for any pair of isomorphic dense maximal pure subgroups, there is an automorphism mapping onto . We show that these classes are extensive and pay attention to...
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