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

By Shalom Feigelstock

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Hence mrs - ss = pr 1s. Now (ss 1 , p) = 1, and so (m,p) = 1. ll i E I) with p a prime such that P ¢ {qi liE 1}, and (m,p) = 1. exists Theorem 4. 6: equivalent: 1) 2) 3) Let G be a rank one torsion free group. G is a strongly semisimple ring group. G is a semisimple ring group. e. G ~ Q+) or k = 0 for infinitely many positive integers n. n 1) • 2): Obvious. Proof: 2) • 3): Corollary 2. 4. If k = ~ for all positive integers n, then the one and only n non-zeroring with additive group G is isomorphic to Q, and so G is a strongly semisimple ring group.

Suppose that H is not nil. Let S be a non-zeroring with s+ = H, and 1et T be the zerori ng with T+ = K. The ring direct sum R = S (f) T satisfies R+ = G, and R2 ~ 0. Since T <1 R, T = · 47 Clearly T = (t1 •••• ,tn), and so T is not nil. The same argument reversing the roles of H and K yields that H is finitely generated. Hence G is finitely generated. 12: Let G be a non-torsion free group. G is a strongly Noetherian ring group if and only if G is finitely generated. 10, it suffices to show that if G is a strongly Noetherian ring group, then G is finitely generated.

1 ~ 0 for some 1 -< i -< n-1. ) > t(g·g ) > t(gnJ = t(G ), a contradiction. , G is an ideal in R. n n_2n-l n 2n- 1 Put R= R/Gn. By the induction hypothesis R 0, or R c G - n 2n 2 . , v(G)~2-l. 8. • ,n. The following are equivalent. t(Gi) is not idempotent for all I < i < n. 2) v(G) ~ n. 4. 48, and so v(G) = m. 2) •1): §2: Gi is the Nilpotence w1thout boundedness conditions, and generalized nilpotence. In the previous section, groups G were considered for which there exists a positive integer n such that Rn = 0 for every (associative) ring R with R+ = G.

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