ABELIAN GROUPS WITH SEMI-LOCAL ENDOMORPHISM RING

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?” ([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.,[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:

	is semi-local if and only if is finitely generated and is a direct sum of form , where is a torsion-free subgroup of such that is semi-local.
	If is divisible, then is semi-local if and only if has finite rank.
	If is torsion-free reduced of finite rank, then is semi-local if and only if , for all but finitely many prime integers .

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.