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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.

This thisideal ideal isisthe the entire entire ring ring I). E I). (m,p) = 1, there exist integers (m,p) there exist integers x,y such such that xm xm ++ py == 1. 1 Let mr Then xm element in pmZ(q:1 1 li E I), xm •• .!!!!. +an I), s E mZ(q:1 li E I). , xm unityniodulo modulo pmZ(qi 1 li E 1), modular modular ideal. I proper ideal ideal in mZ(qi 1 1i E I)~ I) Q, and let let mr be be the Let I~ 00 be be aa proper Q, and 1 smallest positive integer in I c rmZ(qi 1i E 1), I. Clearly I= I), and in I.

1 Let mr Then xm element in pmZ(q:1 1 li E I), xm •• .!!!!. +an I), s E mZ(q:1 li E I). , xm unityniodulo modulo pmZ(qi 1 li E 1), modular modular ideal. I proper ideal ideal in mZ(qi 1 1i E I)~ I) Q, and let let mr be be the Let I~ 00 be be aa proper Q, and 1 smallest positive integer in I c rmZ(qi 1i E 1), I. Clearly I= I), and in I. smallest positive for all (mm, q1) = I. Since I is aa proper ideal in mZ(qi 1 1i E 1), I), (rm, qi) = 1 for all i E I. I. proper ideal I}, and rr t~ 1. 1. Let pp be be aa prime prime such such that pir.

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