Cellular covers of Abelian groups.
Cellular covers of modules and in particular groups have been under intensive investigation recently. For instance every cotorsion-free group can be the kernel of a cellular cover of arbitrary large cardinality by a result due to Buckner and Dugas. Moreover, surjective cellular covers of divisible Abelian groups have been completely characterized by several researchers including the second author. Giving a new and simpler proof of this result, the two authors mainly concentrate in the present paper on reduced Abelian groups. On the one hand several classes of Abelian groups that allow only the trivial cellular covers are listed (e.g. the reduced cotorsion groups). On the other hand, it is shown that even rank one groups may have arbitrarily large cellular covers. In the last section possible kernels of cellular covers are fully described as direct sums of two groups: one is a reduced torsion-free module over the \(\mathbb{Z}\)-adic completion of the integers (whose structure is described) and the other is a cotorsion-free group. The paper is a substantial contribution to the investigation of cellular covers of Abelian groups.
scientific article; zbMATH DE number 5565950
scientific article