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

By Shalom Feigelstock

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G2 = nH(g 1·g 2 J, is a ring K is cyclic, The following are equivalent: Either G is cyclic, or G""' Z(p)@ Z(p), with p a prime. G is a strongly principal ideal ring group. G is an associative strongly principal ideal ring group. 1) • 2): Non-trivial cyclic groups are clearly strongly principal ideal ring groups. Suppose that G = (x 1 ) (f) (x 2 ) with lx;l = p a prime, i = 1,2. Let R be a ring with R+ = G, and R2 '! 0, and let I be a proper ideal in R. Then III = 0, or p, and so I is generated by a single element.

Integers. x = _ _ In particular m2-1 y 1 - k s 1y 1 , and y2 - (k s 2 + m p )y 2 • Hence if m2 > 1, yl y2 y2 m -1 k s 1 = l(mod p), and k s 2 + m p 2 : l(mod p), which implies that yl y2 y2 m -1 p k , and p s 2 • However k s? + m p 2 = O(mod p), so either yl yl "" yl t plky 1 or pls 2 , 44 t a contradiction. 4: ring groups. There are no mixed (associative) strongly principal ideal Proof: Let G be a mixed associative strongly principal ideal ring group. 1, G = H(i) K, H ~ 0, K ~ 0, with H and K both cyclic, or both associative nil.

5: (Wickless [74, pp. 253-254]): Let Gi be a rank one torsion free group with t(Gi) = {i,i, ••. ,i, •.. ro Gi. For every positive integer i, let ei E Gi with 1

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