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Then xy = z+a, z E D, a E (R+)t. Suppose that a ~ 0. Since a E (R+)t• a annihilates D. Let 0 ~ b E (R+)t. There exist x1 , z1 € D such that x = lblx 1 , z = lblz 1 . Therefore ab = [lblx 1y- lblz 1 lb = 0. Similarly ba = 0. Hence a annihilates R, a contradiction. 16: Let R be an (associative) ring with trivial annihilator satisfying the DCC for ideals. Then R is the direct sum of a torsion free ring and a torsion ring. 15. 17: An (associative) ring R possessing only finitely many ideals can be embedded in an (associative) ring with unity possessing only 57 finitely many ideals.

Clearly R5 = 0 for every ring i=l 1 R with R+ = G. •• ), h{e 2 ) = {2,2, ••• ,2, ••• ). and h(e 3) = (4,4, ••. ,4, ••• ). • ) ]. for i+j e. •e. 1 J ~ 3 = otherwise induce an associative ring structure R on G with e~ = e 3 ~ 0. 4. Let ei' i = Hence 1,2,3, be as above and define for i = j < 3 otherwise These products induce a non-associative ring structure S on b with (e 1 ·e 1 ) • (e 1·e 1) = e4 . 6. In this example tne bounds of Webb's Theorem are precisely attained. another such example see [72].

1 there exists an ordinal a; (depending on R;) such that R11 associative ring with that Ra = 0? R+ = ~ i=l G.. = 0, i = l, ••• ,k. Let R be an Does there exist an ordinal a such 1 The concept of a-ni lpotence suggests the following: Definition: Let G be a group. The generalized nilstufe of G, denoted gv(G}, is the first ordinal a such that Ra = 0 for every associative ring R with R+ = G. If no such ordinal exists, put gv(G} = -. Fuchs, [36, problem 94], introduced the following: Definition: Let G be a group.

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