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N Put G = Z(2co) (i) G;· Clearly Rn+l ~ Z(2co) i =1 for every ring R with R+ = G. , N(G) ~ n+l. Choose 0 ~a € Z(2co), and ei € Gi with h(ei) = (co,i ,i, ••• ,i, ••• ). The products r·j ei·ej = for i+j ~ n a for i+j = n 0 for i+j > n for ~ i, j < n induce an associative ring structure R on G. Now n+l e 1 = a ~ 0. Hence n+l = v(G) ~ N(G) ~ n+l, and so v(G) = N(G) = n+l. 4 can be restated as follows: For every group G, v(G) = 1 i f and only i f N(G) = 1. It is not true in general, that v(G) = N(G) for every group G.

A. aj arbitrary cardinals. i = 1, ••• ,m; 56 j = 1 , ••• ,ni . 13: Let G be a group. 10. 14: 1) 2) Let G be a non-nil group. The following are equivalent: Every ring R with R+ = G, and R2 1 0 satisfies the DCC for ideals. m ni . G ""c±)Q+ (±) (±) (±) Z(p~) 0 Z(p~), p. a prime, m, ni 1 1 1 a i=l j=l a. • ,ni. 15: Let R be an (associative) ring with trivial annihilator, such that R+ = D<+l (R+)t• D a divisible group. Then R is a ring direct sum R = S(+) T, with S+ = D, Proof: Let x € D, 0 ~ and T+ = (R+)t.

The proof of this fact may be found in [52, Theorem 11. 49 §3. Descending chain conditions for ideals. 1: G is the additive group of a (left) A group k. ,p. are primes, and are arbitrary cardinals, (associative) Artinian ring if and only if P/k lm, 0 with where a,l3 J 1 m is a fixed positive integer. R R. Hence there exists a positive integer n > k. Put m = k! ) @ (£) Z(p00 ) . a p a prime ap divisible, and so boundedjandso for all n:;, ... R ::J 2! R ::J 3! R be an Artinian ring. Clearly (mR)+ Clearly, R+""(±)Q+(±) (£) ~ Z(p00)(f{i~)Z(p~).

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