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2 Fundamenta Mathematicae

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Almost-free E(R)-algebras and E(A,R)-modules

100%

Göbel R. , Strüngmann L.

Fundamenta Mathematicae

2001

tom 169

nr 2

175-192

Let R be a unital commutative ring and A a unital R-algebra. We introduce the category of E(A,R)-modules which is a natural extension of the category of E-modules. The properties of E(A,R)-modules are studied; in particular we consider the subclass of E(R)-algebras. This subclass is of special interest since it coincides with the class of E-rings in the case R = ℤ. Assuming diamond ⋄, almost-free E(R)-algebras of cardinality κ are constructed for any regular non-weakly compact cardinal κ > ℵ ₀ and suitable R. The set-theoretic hypothesis can be weakened.

On localizations of torsion abelian groups

80%

Rodríguez J. , Scherer J. , Strüngmann L.

Fundamenta Mathematicae

2004

tom 183

nr 2

123-138

As is well known, torsion abelian groups are not preserved by localization functors. However, Libman proved that the cardinality of LT is bounded by $|T|^{ℵ₀}$ whenever T is torsion abelian and L is a localization functor. In this paper we study localizations of torsion abelian groups and investigate new examples. In particular we prove that the structure of LT is determined by the structure of the localization of the primary components of T in many cases. Furthermore, we completely characterize the relationship between localizations of abelian p-groups and their basic subgroups.