Additive Groups of Rings by S. Feigelstock

By S. Feigelstock

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Let R R R(1) = R(l) R,_ and define and for for every every positive positive integer kk inductively define = R, be the the subring subring of R generated form R(k+l) to be generated by by elements elementsof of the form R R(k)+ ~ Q+. Now {ax, vk == R(k)+ Now Vk Vk is aa vector vector x,y E R(k)}. Put Vk (ax, ybl a, bbE R, R, positive integer every positive integer k, and dim space space over QQ for every dimQV1 0v1 == n. The terminates at 0, and has length length at descending descending chain V1 v1 ~ vV2 ...

A? For a an an arbitrary ordinal, ordinal,does does there there exist existaagroup group G G 35 4 Other Otherring ring properties properties §1. Semisimple, prime, semiprime, simple, division ring, field, radical ring. Definition: Let w be be aa ring property: property: A said to to be be an an A group GG is said 71 (associative) 71—group there exists an such that (associative} w-group ifif there an (associative) (associative}iT—ring n-ring RR such If GG is not R+ = G. G. If not (associative) (associative} nil, nil,and and every every (associative) (associative} ring R R 2 satisfying R2 R ~ 0, and R+ = GG is aa71—ring, n-ring, then then G is said said to tobe be an an G (associative) strongly (associative} stronglyit—group.

And t(G11)) = For -3 -3 [(4,4,... )]. • ) ]. Put G == G G.. ), R with R+ == G. G. ), h(e3) (4,4,... ). The products h{e ••• ,2, ••• ). and h(e ••. ,4, ••• ). The 3) ==(4,4, 2 ) =={2,2, for u-i i+j = e. •e. •e. 1 J 1 3 ~ 3 = 1 0 otherwise induce an anassociative associative ring ring structure R R G with e~ = e3 induce on G e 3 ~ 0. 4. •e. (. e1) = e4. S4 N(G) = ) (e 1 ·e 1 ) • (e ·e = e . 6. 1 1 4 Theoremare areprecisely precisely attained. attained. In this thisexample example the tne bounds bounds of of Webb's Webb's Theorem another such see [72].

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