Vol. 53, No. 2, 1974

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Strongly semisimple abelian groups

Ross A. Beaumont and Donald Lawver

Vol. 53 (1974), No. 2, 327–336

For an abelian group G and a ring R,R is a ring on G if the additive group of R is isomorphic to G. G is nil if the only ring R on G is the zero ring, R2 = {0}.G is radical if there is a nonzero ring on G that is radical in the Jacobson sense. Otherwise, G is antiradical. G is semisimple if there is some (Jacobson) semisimple ring on G, and G is strongly semisimple if G is nonnil and every nonzero ring on G is semisimple. It is shown that the only strongly semisimple torsion groups are cyclic of prime order, and that no mixed group is strongly semisimple. The torsion free rank one strongly semisimple groups are characterized in terms of their type, and it is shown that the strongly semisimple and antiradical rank one groups coincide. For torsion free groups it is shown that the property of being strongly semisimple is invariant under quasi-isomorphism and that a strongly semisimple group is strongly indecomposable. Further, for a strongly indecomposable torsion free group G of finite rank, the following are equivalent: (a) G is semisimple, (b) G is strongly semisimple, (c) GR+ where R is a full subring of an algebraic number field K such that [K,Q] = rank G where Q is the field of rational numbers and R = Jπ, where π is either empty or an infinite set of primes in K,(d)G is nonnil and antiradical.

Mathematical Subject Classification 2000
Primary: 20K99
Received: 15 May 1973
Published: 1 August 1974
Ross A. Beaumont
Donald Lawver