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By Dr. Leslie Cohn (auth.)

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3. li. is such that e x(H(~) ) is a polynomial function; and suppose that e ~(H(~)) = ~ji(B)bi i=l (bill) 35 is the factorization of e k'H'-''( ( ~ into irreducible polynomial functions. ,a). ;a and Z ~ ~ , Remark. (Z e ~ , ~ £ ~). Ji divides q(Z)J i. Let Ji(~) and Cj. be as in the preceeding corollary. (VI~) = 0 for V E ~ M ' ~ e N. (~) = I when ~ = e. ,a) such that p(m)Ji(~) = Xi(m)Js(m)(i)(~) for all ~ e ~ and m e ~ . 0(m)Ji(~) = Js(m)i(~). m + ~m)Ji({) Letting ~ = e, we see that ~(m) m i; hence, It then follows that the map ~ + ~ sending (5 E H fixed) assumes only finitely many values, hence, 36 being continuous, is identically equal to Ji(~) on ~ , then p(V)J i = O for V e ~ .

Or simply a graded ring) if for subgroup R I of R such that A) and . R 1 Suppose again that R is a rin£ and that A is an additive Suppose also that A is partially ordered by a relation lI < 12, then 11 + ~l < 12 + ~ (I 1 , 12, p A-filtered ring if we have additive i) RIR~ 2) R I -~ R p if I < ~, and 3) U R I leA R I+~ (I, ~ e s A). semi~ro~p. < such that if We say that R is a subgroups R I of R such that A), = R. If R is graded by the subspaces R I ring by means of the subspaces R l = ~ @ (I e A), we obtain a filtered P of R° Similarly, if R is a filtered ring, we obtain a graded ring G(R) in the followinR ray.

3. If XI, X2 e~ , then q(XI)F(2)(X2) - q(X2)F(2)(XI) + [F(2)(XI), F(2)(X2)] = F(2)([XI,X2]). xz' ×2 ~ ' q(xz)F(2)(~I~)(x2)= - X B(×I,XB~)~(X2'[x-B'vj]~)vJ = - [B(X2~'I , [~2(XI~-I),Vj])Vj = ~([~2(XZ~-I),X2~'Z]) = w([~2(Xl['l),~I(X2['I)]). Hence, 45 q(XI)F(2)(~I~)(X2) - ~(X2)F(2)(~I~)(XI) + [F(2)(~I~)(XI) , F(2)(vI~)(X2 )] = ~([~2(XI{-I), ~l(Xl{'l)] . ~I(XI[-I)] + [~(XI~'I), X2~-I)] ) = ~([XI~-I,x2~-I]) = F(2)(vI~)([XI,X2]), as claimed. b. If XI, X2 e ~ l , then q(Xl)¢i(x2) - Q(x2)~i(xi) = ~/[Xl,X2]).

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