By S. Feigelstock
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N. 4. 4. 1) 30 30 I i 1): 2) •1): If If t(Gi) is idempotent idempotent for forsome some 1 ~< i ~ < n, then additive group group of aa subring subring of Q, and so so v(G) v(G) == m. 48, and §2: 1 -I Gi is the the Fiilpotence w1thout without boundedness conditions, and and generalized generalized nilpotence. nilpotence. Nilpotence boundedness conditions, were considered consideredfor for which which there there exists In the the previous previous section, section,groups groups G were G Rn R aa positive integer integer nn such such that Rn = every (associative) (associative) ring ring R = 00 for every with R+ = G.
E1) = e4. S4 N(G) = ) (e 1 ·e 1 ) • (e ·e = e . 6. 1 1 4 Theoremare areprecisely precisely attained. attained. In this thisexample example the tne bounds bounds of of Webb's Webb's Theorem another such see . . 8: For G satisfying v(G) A group G N(G) == ~. v(G) << N(G) be aa rank one torsion torsion free group iI == 11,2, ,2, let G. )], ,0, 1 ,0, ••. )]. l ~. ~. 1 , ••••~. 1 , ••. ). G2. h(e ••••~,l, ••• ). Put G = G1 ~ G easily verified verified 2) == l~,l,~,l, 2 • It For 28 v(G) = 2. that v(G) The products The [0 = j 2 for i=j=2 for i = e2 otherwise induce aa non-associative non—associativering ring structure structure R on G.
B) is finite, HH an G == B t:t) H, B For For the the associative case case consider consider G B finite, associative quasi-nil quasi-nil torsion associative torsion free free group. group. Let lSI = m. m. The The ring by products products of elements multiplications multiplications on on G 6 are are determined determined by elements in B, for which thereare areclearly clearly only only finitely finitely many which there many possibilities, possibilities,products productsofofelements elements in B B and products products of of elements elements in H.
Additive Groups of Rings by S. Feigelstock