By S. Feigelstock
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Extra resources for Additive Groups of Rings
6 the direct one torsion free By Corollary directsum sum of nn rank rank one freegroups groups has has strong nilstufe nilstufe <~ 2n-l or co. ) G1. Gi. Then v(G) ~ 2" - 1. 1. 2. Let n >> 1, and the theorem is true, 1, supposethe thetheorem theorem truefor for the the direct su~pose is istrue direct sum sum of less less than than nn homogeneous If t(Gi) == t(GjJ for 1 ~< i ~ jj <~ n, groups n, then groups of non-idempotent non-idempotent type. If G groupsof of non-idempotent G is the the direct directsum sum of of less less than than nn homogeneous homogeneous groups non-idempotent v(G) << 2n-l —- 1 by type, and and v(G) the induction induction hypothesis.
Clearly 0D is an ideal ideal in R, and R = RID Therefore there there exists exists aa an R/D is nilpotent. nilpotent. Therefore R2n 2 2 = 0, positive integer integer nn such D. Hence R n cc o such that Rn = 0, or Rn c D. 1. 1. 1 (associative) (associative) ring structures structures to to the the torsion torsion free free case. case. 2: (associative) (associative) ring ring be aa torsion torsion free G Let G be free group group with r(G) r(G) << if and onlyifif R with R+ == GG is nilpotent nilpotent if and only R m.
Then not torsion free. 4, and 1 ~< kk << ~. 3, and and so so v(G) v(G) == ~. aa contradi cti on. contradiction. 10: Let G be aa mixed mixed group. group. v(G) < ~ if if and and only G D divisible torsion G == D~ H, with D aa divisible torsiongroup, group, and and HH aa torsion free free group group with v(H) < ~. v(G) << ~. 9. 9. Theorem D divisible torsion Conversely, suppose that G aa divisible torsion group, group, and Conversely, suppose G == D(f) H, D H v(H) = nn < ~. Let R R be an an associative associative ring H aa torsion free free group group with with v(HJ be ideal in R, and (RIO)+~ H.
Additive Groups of Rings by S. Feigelstock