By M. Sion

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X)s Define to Cauchy. and work A(x) = with ~ i

Is a Y-ring. if the m a p If R is a p r e - Y - r i n g , Y-ring if and only Y(r) (yl(r),y2(r) .... ), is a h o m o m o r p h i s m = let R w be the set of then R is a Y:R---~ R w, d e f i n e d b y of '~-rings. D Proof: Clear. Note that since 1 is the i d e n t i t y map, the m a p Y:R ~ R ~ is one-one. , following p. pre-~-ring. Sul]pose t h e r e are o p e r a t i o n s co ~d logkt(x) = ~ - ( - 1 ) n y n + l ( x ) t n, a l l x£R. n=l , we can c a l c u l a t e l(x) 1 , I~ y2 n. k n(x) = de yl (x) (x) yl(xI n(x) T h e n we s u p p o s e that b y the d e t e r m i n a n t so t h a t kn(x) that this for each n > i, and each x 6 R the e l e m e n t on the right h a n d is d e f i n e d .

Put a rather on W R. M:WR----~ R ~ d e f i n e d by n/d M ( ( w ! , w 2 .... )) = Thus (rl,r 2 .... ) where r = n ~dw dh d can identify rI = w 1 2 r2 = w I + 2w 2 3 r3 = wI = r4 + 3w 3 4 wI + 2w22 + 4w 4 etc. If R is t o r s i o n - f r e e , M is a one-one map and we W R with its image Proposition: there M(WR) WR(M) c R . , under G. w i t h sum and product integer coefficients l F n depends [wl I i d i v i d e s on t w o n]) sets of variables: Indeed such j M ( ( W l , w 2 .... ) ) + M ( ( w l , w 2 ....