Additive Groups of Rings by S. Feigelstock

By S. Feigelstock

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The products h{e ••• ,2, ••• ). and h(e ••. ,4, ••• ). The 3) ==(4,4, 2 ) =={2,2, for u-i i+j = e. •e. •e. 1 J 1 3 ~ 3 = 1 0 otherwise induce an anassociative associative ring ring structure R R G with e~ = e3 induce on G e 3 ~ 0. 4. •e. (. 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 [72]. [72]. 8: For G satisfying v(G) A group G N(G) == ~.

13: Let GG be and onlyif if GP is aa radical and only radicalring ringgroup groupfor forsome some prime prime p. p• . 7. 7. 1 Conversely, suppose that radical ring ring with with R+ = G, and R2 ~ 0. Conversely, suppose that RR is aa radical R = R Then RR is a ring direct sum, R ~. Rp•, with R; = GGP for every every p p 22 ppprime pr1me 22 such that Rp ~ 0. Now there exists aa prime R prime p. , Gp is aa radical radical ring group. group. 13 reduces thethe study of of torsion groups to the p-primary case, case, pp a prime.

1 ~ 0 for some 1 -< i -< n-1. 2, t(G > Consequence > t(g·g ) > t(gnJ = t(G ), aa contradiction. ) 1 n n an ideal in R. , G c G and similarly G G is an ideal R. n nnn n_22n—1 n-l n 2n—1 2n- 1 R Put R== R/Gn. By the induction hypothesis RR 0, or R c G - n 2n 2 . 2,andso R =Gn=O, v(G)~2-l. 8. 8. 13: Let G G; , wi th Gi aa rank G == _@ ® G1, 1 =1 are equiva1ent. equivalent. • ,n. ,n. n. for all I < i < n. t(Gi) is not not idempotent idempotent for 2) v(G) v(G) < n. ~ 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.

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