By S. Feigelstock
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Extra resources for Additive Groups of Rings
T-nilpotent. Fromnow nowononT-nilpotence 1—ni ipotence meant left T-nilpotence. 1-nilpotence. 8: Every Every associative associative ring G be Let G be a group. group. R with R+ = GG R is 1—nilpotent onlyif if G T-nilpotent ifif and and only with D aa divisible torsion group, group, divisible torsion G == D@H D®H with D torsion free free group such that that every and HH aa reduced reduced torsion group such every associative associative ring ring SS + with Ss+ = H with H is 1-nilpotent. T-nilpotent.
W-group. The above aboveterminology terminologywas wasfirst first employed by Beaumont andLawver, Lawyer,(6], , in in The employed by Beaumont and studying semisimple, and andthe the associative associative strongly studying the the associative associative Jacobson Jacobson semisimple, Jacobson semisimple groups. Jacobson semisimple Throughoutthe theremainder remainder thissection, section, the the rings considered will be Throughout ofofthis considered will be assumedtotobe be associative. associative.
That xm Supposethat that m >> n+l. Then Thenthere there exist exist integers 0. ,amll i =1 m-1 m-2 m-2 z I: 1 i aix . ,m—2, j' 0, and aml m- 1 a xm-1 == o. amlxm_I G is torsion torsi on free free xm-1 = = 0, aa contradiction. 0. Since G m-1 m-2 m-2 i+l < m-2. x for some < Ii ~ Therefore a; ~ 00 for some ~ m-2. Now m-1 i=l 1 i=l . 1 m-3 . x 1 + •. Repeating m-3 Repeating the above more above procedure procedure more time m-2 . 1 1= 1 am_lxm_I -= —am-lx i =1 1 1 1 yields that aa1xm_I a 1 ~a. Hence xm-l == 0, aa contradiction.
Additive Groups of Rings by S. Feigelstock