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By S. Feigelstock

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G}. Gardner,[41, [41, p. p. 386], annihilator Gardner, 386], introduced introduced the the following following ascending ascending annihilator series for G(l) == the annihilator of G. For series for aagroup group G: G: Put G(l} the absolute absolute annihilator 34 every G/G(cL) every ordinal a, put G/G(a) define G(s) = u G(a). 12: Let then gv(G) gv (G) ~ a+ 1. (1) (1) = G(a+l)/G(a). G G be a group. If GG == G(a) For J3 aa limit ordinal, ordinal, for some ordinal for some a, Proof: ~: Let RR be an associative associative ring with be an with R+ = We wish show that We wish to to show = G.

Andwill will be be G. ,k. ••.. ,k. Then + kk associative ring RR with RR+ == (f) G. G. is nhlpotent. nilpotent. ,k. )<~. 2. 2. 3 vv(G1) ( Gi ) << ~. 4 is obviously necessary. The same sameisis true true for necessary. 3. (Wickless (Wickless [74, [74, pp. ) be rank one free group group with t(Gi) == {i,i, ••. ,i, •.. ) , G1 and let G G1. ro and let Gi. ). h(ei) = = (i,i, ... • ). induce an an associative The products e1•e3 The ei·ej == ei+j induce R+ ring structure R, with R+ = ring G. For For every integer n, every positive positive integer = G.

G. Theequivalence equivalenceofof 1) 1) and and 3) 3) as as well as The as aa general general treatment treatment ofofsemisimple semisimple and strongly strongly semisimple ring groups maybebefound foundinin [6). [6]. 3: equivalent: Let G be free group. group. G be aa torsion free 1) G G is aa field fieldgroup. group. 2) G is aa division division ring ring group. group. G 3) G is aa simple simple ring ring group group G The following following are The are an arbitrary cardinal G is divisible. e •• G ~ (i) Q+.

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