Kulikov's problem on universal torsion-free abelian groups revisited

DOI10.1112/BLMS/BDR055zbMATH Open1237.20051arXivmath/0112253OpenAlexW3099705433MaRDI QIDQ3104000

Saharon Shelah, Lutz Strüngmann

Publication date: 19 December 2011

Published in: Bulletin of the London Mathematical Society (Search for Journal in Brave)

Abstract: Let T be an abelian group and lambda an uncountable regular cardinal. We consider the question of whether there is a lambda-universal group G^* among all torsion-free abelian groups G of cardinality less than or equal to lambda satisfying Ext(G,T)=0. Here G^* is said to be lambda-universal for T if, whenever a torsion-free abelian group G of cardinality less than or equal to lambda satisfies Ext(G,T)=0, then there is an embedding of G into G^*. For large classes of abelian groups T and cardinals lambda it is shown that the answer is consistently no. In particular, for T torsion, this solves a problem of Kulikov.

Full work available at URL: https://arxiv.org/abs/math/0112253

zbMATH Keywords

torsion-free Abelian groupsExt-functoruniversal groupstorsion Abelian groupsKulikov problem

Mathematics Subject Classification ID

Torsion-free groups, infinite rank (20K20) Extensions of abelian groups (20K35) Homological and categorical methods for abelian groups (20K40) Torsion groups, primary groups and generalized primary groups (20K10)

Related Items (2)

Non-existence of universal members in classes of Abelian groups. ⋮ KULIKOV'S PROBLEM ON UNIVERSAL TORSION-FREE ABELIAN GROUPS

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