Additive Groups of Rings (Chapman & Hall CRC Research Notes - download pdf or read online

By Shalom Feigelstock

ISBN-10: 0273085913

ISBN-13: 9780273085911

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Extra resources for Additive Groups of Rings (Chapman & Hall CRC Research Notes in Mathematics Series)

Example text

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. In this section the bound on the degree of nilpotence will be deleted.

If no such positive integer n exists, then v(G) = ~. The strong nilstufe of G, N(G) is defined as above, with the associativity of R deleted. Clearly the (associative) nil groups are precisely the groups satisfying (v(G) = 1) N(G) = 1. 1: For every positive integer n, group G satisfying v(G) = N(G) = n. there exists a torsion free For every positive integer i, let G. •• )]. • Clearly Rn+l =0 forevery i=l , ring R with R+ = G. •• ,i, ••• ). +. , J for i+j -< n 0 for i+j > n n induce an associative ring structure R on G.

Therefore a; ~ 0 for some . , a xmm-2 = ~ i ~ m-2. x m-1 i=l 1 m-3 . x 1 + • Repeating the above procedure m-3 more time . 1 1= 1 yields that a1xm-l = 0, with a 1 ~a. Hence xm-l = 0, a contradiction. Therefore xn+l = 0 for all x € R. 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.

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Additive Groups of Rings (Chapman & Hall CRC Research Notes in Mathematics Series) by Shalom Feigelstock


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