ABSTRACT
The abelian groups with semi-local endomorphism ring are characterized, with one exception: the infinite rank torsion-free ones.
A ring is called semi-local if
is semisimple artinian.
In his authoritative book on infinite abelian groups, Laszlo Fuchs, the leading expert on this topic, asks: “For which abelian groups is the endomorphism ring semi-local?” (Citation[1], Problem 84). This problem has been open for 26 years. Modules whose endomorphism ring is semi-local have been investigated by several authors (see, e.g.,Citation[3] and the literature listed there).
The main result of this paper is the following theorem:
Theorem.
Let
be an abelian group with endomorphism ring
and torsion subgroup
. Then:
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If
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If
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ACKNOWLEDGMENTS
The author expresses his gratitude to Laszlo Fuchs for his kind help and suggestions, to John Dauns for valuable ring-theoretic help and to Tulane University for its warm hospitality.
While writing this article, the author was supported by a Research Fulbright Fellowship (number 22836) and was on leave from the Department of Mathematics and Computer Science, “Babeş-Bolyai” University, Cluj-Napoca, Romania.