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Co-Hopfian and boundedly endo-rigid mixed abelian groups

Mohsen Asgharzadeh, Mohammad Golshani and Saharon Shelah

Vol. 327 (2023), No. 2, 183–232
Abstract

For a given cardinal λ and a torsion abelian group K of cardinality less than λ, we present, under some mild conditions (for example, λ = λ0), boundedly endo-rigid abelian group G of cardinality λ with tor (G) = K. Essentially, we give a complete characterization of such pairs (K,λ). Among other things, we use a twofold version of the black box. We present an application of the construction of boundedly endo-rigid abelian groups. Namely, we turn to the existence problem of co-Hopfian abelian groups of a given size, and present some new classes of them, mainly in the case of mixed abelian groups. In particular, we give useful criteria to detect when a boundedly endo-rigid abelian group is co-Hopfian and completely determine cardinals λ > 20 for which there is a co-Hopfian abelian group of size λ.

Keywords
black boxes, bounded endomorphisms, co-Hopfian groups, endomorphism algebras, mixed abelian groups, $p$-groups, set theoretical methods in algebra
Mathematical Subject Classification
Primary: 03E75, 16S50, 20K30
Milestones
Received: 23 February 2023
Revised: 8 October 2023
Accepted: 6 November 2023
Published: 12 March 2024
Authors
Mohsen Asgharzadeh
Hakimiyeh
Tehran
Iran
Mohammad Golshani
Institute for Research in Fundamental Sciences (IPM)
Tehran
Iran
Saharon Shelah
Einstein Institute of Mathematics
The Hebrew University of Jerusalem
Jerusalem
Israel
Department of Mathematics
Rutgers University
New Brunswick, NJ
United States

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