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Torsion-free E-uniserial groups of infinite rank. (English) Zbl 0693.20051

Abelian group theory, Proc. 4th Conf., Perth/Aust. 1987, Contemp. Math. 87, 181-189 (1989).
[For the entire collection see Zbl 0666.00004.]
Let G denote a torsion-free abelian group, E its endomorphism ring and C the center of E. Considerable research has been devoted to the structure of G as an E-module. In particular, G is called: strongly homogeneous if Aut G, the group of automorphisms of G, acts transitively on the pure rank one subgroups of G; E-uniserial if the lattice of E-submodules of G is totally ordered; and E-transitive if E acts transitively on the pure rank one subgroups of G. Some very strong results relating these properties have been obtained by Hausen, Reid and others in the case where G is of finite rank. The note under review investigates the extension of the finite rank results to infinite rank.
Call G strongly C-homogeneous (respectively, E-transitive with respect to C) if G is a torsion-free C-module and Aut G (respectively, E) acts transitively on pure rank one C-submodules of G. The torsion-free assumption forces C to be an integral domain. A fully invariant subgroup H of G is called C-scalar if \(H=Gc\) for some \(c\in C.\)
Theorem 1.1. The following are equivalent. (1) G is reduced, E-uniserial and every fully invariant subgroup is C-scalar. (2) G is E-transitive with respect to C, E-cyclic, and C is a discrete valuation domain.
Theorem 1.3. Suppose that G is a torsion-free separable C-module. The following are equivalent. (1) G is reduced E-uniserial and every fully invariant subgroup is C-scalar. (2) G is E-transitive with respect to C and C is a discrete valuation domain. (3) G is strongly C-homogeneous and C is a discrete valuation domain.
An additional theorem gives sufficient conditions for a reduced E- uniserial G to be a separable C-module, and three interesting open questions are posed.
Reviewer: C.Vinsonhaler

MSC:

20K20 Torsion-free groups, infinite rank
16S50 Endomorphism rings; matrix rings
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups

Citations:

Zbl 0666.00004