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Abstract
The non-trivial hereditary monocoreflective subcategories of the Abelian groups are the following ones: {G ∈ Ob Ab | G is a torsion group, and for all g ∈ G the exponent of any prime p in the prime factorization of o(g) is at most E(p)}, where E(·) is an arbitrary function from the prime numbers to {0, 1, 2, …,∞}. (o(·) means the order of an element, and n ≤ ∞ means n < ∞.) This result is dualized to the category of compact Hausdorff Abelian groups (the respective subcategories are {G ∈ Ob CompAb | G has a neighbourhood subbase {G α } at 0, consisting of open subgroups, such that G/G α is cyclic, of order like o(g) above}), and is generalized to categories of unitary R-modules for R an integral domain that is a principal ideal domain. For general rings R with 1, an analogous theorem holds, where the hereditary monocoreflective subcategories of unitary left R-modules are described with the help of filters L in the lattice of the left ideals of the ring R. These subcategories consist of those left R-modules, for which the annihilators of all elements belong to L. If R is commutative, then this correspondence between these subcategories and these filters L is bijective.
An abelian p-group G has a nice basis if it is the ascending union of a sequence of nice subgroups, each of which is a direct sum of cyclic groups. It is shown that if G is any group, then G ⊕ D has a nice basis, where D is the divisible hull of p ω G. This leads to a consideration of the nice basis rank of G, i.e., the smallest rank of a divisible group D such that G ⊕ D has a nice basis. This concept is used to show that there exist a reduced group G and a non-reduced group H, both without a nice basis, such that G ⊕ H has a nice basis
An affirmative answer is given to a problem on factoring finite abelian groups by subsets in a particular case.
Abstract
This paper gives a complete description of the behaviour of torsion-free abelian groups of rank 3 with respect to endoprimality.
We prove that every Abelian group G is determined up to an isomorphism by the subgroup lattice of the group ℤ × G and some other similar results.
