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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On endomorphisms of abelian topological groups

Authors: Eli Katz and Sidney A. Morris
Journal: Proc. Amer. Math. Soc. 85 (1982), 181-183
MSC: Primary 22B05; Secondary 16A65
MathSciNet review: 652437
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Abstract: A family $ \Phi $ of continuous endomorphisms of a topological group $ G$ is said to be small if for every subgroup $ H$ of $ G$ of cardinality $ {\text{card}}(H) < {\text{card}}(G)$ there exists an element $ g \in G$ such that $ \Phi g \cap H = \emptyset$. M. I. Kabenjuk [5] proved that if $ G$ is a compact connected Hausdorff abelian group of countable weight then every countable family $ \Phi $ of nontrivial endomorphisms of $ G$ is small. He asked if "compact" can be replaced by "complete". In this note the answer is given in the negative, but it is shown that "compact" can be replaced by "locally compact".

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Keywords: Endomorphism ring, topological group, abelian group, representable ring
Article copyright: © Copyright 1982 American Mathematical Society