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**Sample text**

T4ult G ""Hom{~. G) ... Hom{G~. B)(+)Hom{H~. H), since Hom(X,Y) = 0 for X a torsion group, and Y a torsion free group. If G is quasi-nil then so is H. Therefore Hom{H~. H) ""'r~ult H determines only finitely many non-isomorphic ring structures on G. Hom{G~. B) ""'Hom(B~B. B) t+) Hom{H~, B) (f) Hom(B~, B) (t) Hom{Hfti, B). B, B~ are finite, and so the first three summands above are finite. 3, and B is finite, Hom{H·~. B) is finite. For the associative case consider G = B t:t) H, B finite, H an associative quasi-nil torsion free group.

Z(p~) is isomorphic to the additive group of p-adic numbers. If u1(x,y) and u2(x,y) are two ring multiplications on G with u1 a non-zero multiplication, then u2(x,y) = wu 1(x,y), w a p-adic number, for all x1,x2 € G. Multiplication by p is an automorphism of Z(p=). Therefore, the multiplications u1(x,y) and p- 2Pu 1(x,y), p a p-adic unit, define isomorphic ring structures on G. Every non-zero p-adic number is of the form pkw, k an integer, w a p-adic unit. Therefore the elements of Hom(~. Z(p=)) induce three non-isomorphic ring multiplications on G, the zero multiplication, u1(x,y), and Pu 1(x,y).

3, and so v(G) = ~. a contradiction. 10: Let G be a mixed group. v(G) < ~ if and only if G = D~ H, with D a divisible torsion group, and H a torsion free group with v(H) < ~. Proof: If v(G) < ~. 9. Conversely, suppose that G = D(f) H, D a divisible torsion group, and H a torsion free group with v(HJ = n < ~. Let R be an associative ring with H+ = G. Clearly u is an ideal in R, and (RIO)+~ H. Hence (R/D)n+l = 0, or Rn+l =D. , v(G) ~2n+l. The problem of classifying the groups with finite nilstufe therefore reduces to the torsion free case.