Download e-book for kindle: Additive Groups of Rings (Chapman & Hall CRC Research Notes by Shalom Feigelstock

By Shalom Feigelstock

ISBN-10: 0273085913

ISBN-13: 9780273085911

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

Example text

Ring group for every group H. i:)H is a radical Proof: Let S be a radical ring satisfying s2 i O, and s+ = G. Let T be the zeroring with T+ = H. The ring direct sum R = S (f) T, satisfies R2 iO, R+=G~H, and JlR)=R. 8: If G possesses a subgroup isomorphic to Q+ ~ Q+ , G is a radical ring group. 10. 7, it suffices to show that Q+ (1) Q+ is a radical ring group. Let a,b be independent elements in Q+@ Q+. The products a2 = b, ab = ba = b2 = 0 induce a ri11g structure R on Q+ ~ Q+ satisfying R2 '! 0, but R3 = 0.

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.

N. such that H pi 4 (tj j=l a. Z(p~), cxJ. a cardinal number, j = 1 , ••• ,ni. J Clearly G is of the form of condition 2). 1. 8: Let G be a torsion group. G is the additive group of a ring with trivial annihilator satisfying the DCC for ideals if and only if G sati~fies one, and hence all, of the equivalent conditions in Theorem 4. 5. 1. 7. 4, is a ring with trivial annihilator satisfying the DCC for ideals. 9: Let G be a non-nil torsion group. The following are equivalent: · 1) Every (associative) ring R with R+ = G, and R2 ~ 0 satisfies the DCC for ideals.

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


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