Additive Groups of Rings (Chapman & Hall CRC Research Notes by Shalom Feigelstock

By Shalom Feigelstock

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Suppose that G is of the form 2) with H nil. As in the case of form 1) it suffices to investigate Hom(~. G). Arguments similar to those used (±) Hom(~, D } in case l} reduce the investigation to Hom(H~, D) ""' p prime P Since Dp = 0 for all but finitely many primes p, it suffices to consider Hom(H~, DP), p a fixed prime, with OP ~ 0. Now r(HSltt) = 1, and so 23 every ~ € Hom(HAH, Dp) is a map ~= ~H ~ Z(p=). Changing the summand Z(p=) only alters the ring determined by ~ by a ring isomorphism. Therefore it suffices to examine Hom(H~.

17. Much additional information about {strongly) semisimple ring groups may be found in [6]. , the additive group of a ring R satisfying R2 i 0, and J(R) = R. These groups have been studied by F. Haimo [44]. Some of his results, and others, will be presented here. 7: Let G be a radical ring group. ring group for every group H. i:)H is a radical Proof: Let S be a radical ring satisfying s2 i O, and s+ = G. Let T be the zeroring with T+ = H. The ring direct sum R = S (f) T, satisfies R2 iO, R+=G~H, and JlR)=R.

V( Gi ) < ~. 4 is obviously necessary. 3. 5: (Wickless [74, pp. 253-254]): Let Gi be a rank one torsion free group with t(Gi) = {i,i, ••. ,i, •.. ro Gi. For every positive integer i, let ei E Gi with 1

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