Additive Groups of Rings (Chapman & Hall CRC Research Notes by Shalom Feigelstock

By Shalom Feigelstock

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11, both H and K are nil. 1 it may be assumed that H is a torsion group, and that K is torsion free. Let R be a Noetherian ring with R+ = G. Since ti <2 R, H = . Let n = k rr lhil. i =1 Then nH = 0. 1, H is divisible, a contradiction. 13: Let G be a strongly Noetherian ring group. and G/Gt are strongly Noetherian ring groups. Then Gt i··, Proof: If G is torsion free, the statement of the corollary is trivial. 10. , Q+. 14: Let G be a strongly Noetherian ring group. either indecomposable, or finitely generated.

By the Nagata-Higman Theorem 26 [48, p. 274], R is nilpotent. Let d be the smallest positive integer satisfying Rd = 0. For every positive integer k, let (O:Rk) = {x E Gl xy = 0 for all y E Rk}. •. < (O:Rd-l) = G is properly ascending. d ~ n+l, Hence [36, vol. 1, p. 120, Ex. 8 (b)]. r(G) = n < 3. 4: Let G be a torsi on free group with following are equivalent: Corollar~ 1) Every associative ring 2) Every associative ring 3) \I(G) ~ =. The R with R+ = G is nilpotent. R with R+ = G is ni 1. n. Proof: The implications 1) • 2) and 3) • 1) are obvious.

13: Let G be a torsion group. G is a radical ring group if and only if GP is a radical ring group for some prime p • . 7. 1 Conversely, suppose that R is a radical ring with R+ = G, and R2 ~ 0. Then R is a ring direct sum, R = ~. Rp• with R; = GP for every 2 p pr1me 2 prime p. Since R ~ 0, there exists a prime p such that Rp ~ 0. , Gp is a radical ring group. 13 reduces the study of torsion radical ring groups to the p-primary case, p a prime. 14: Let G be a p-primary group, p a prime. G is a radical ring group if and only if G t Z(p), and G is not divisible.

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