Jacobson radicals, abelian -groups and the -topology
Patrick Keef
Whitman College, Walla Walla, USA
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Abstract
In studying the Jacobson radical of the endomorphism ring of a separable abelian -group, Sands (1984) identified the useful Condition (C). Our central result is that the group satisfies Condition (C) precisely when it is complete in its -topology, which uses the set of subgroups such that is a direct sum of cyclics as a neighborhood base of 0. This equivalence is then used to compute the Jacobson radicals of the endomorphism rings of a variety of such groups, including those in the so-called Keef class. It is shown that Sands’ attempt to “complete” an arbitrary group with respect to Condition (C) is equivalent to D’Este’s (1980) attempt to show that the -topology is completable. Since Mader (1983) provided a counterexample to D’Este’s result, it follows that Sands’ result also fails.
Cite this article
Patrick Keef, Jacobson radicals, abelian -groups and the -topology. Rend. Sem. Mat. Univ. Padova (2024), published online first
DOI 10.4171/RSMUP/162