By J. P. Levine

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B = Kk(A) constructed to those to c h e c k are above, induced by isomorphic so that the the in- B + A. It is c l e a r that B. = A. 1 1 for i < k, B. 1 = 0 for i > k, -- 26 so that we h a v e isomorphisms C. ~ B. 1 Furthermore i, we Bi = 0 observe isomorphic for that i > k. to the p u s h - o u t the d e s i r e d Proceeding B k-I Bk_ 1 with properties. 1 and if by d o w n w a r d £ < k - I, induction then on is B£ of A ~ ÷ A z+I B£+I This follows from the commutative diagram 0 ÷ B~+ 1 + B~ ÷ B ~ + B ~+I + 0 0 + A£+ 1 ÷ A~ ÷ A ~ + A £+I ÷ 0 with phisms.

Ym } ~ A = It is easy it s u f f i c e s pA = O, since, Ym(1) completely satisfying choose and yi/Yi+l to them as the free m o d u l e s then free m o d u l e This divisor K, F ) - i, Yi(1) A = (k e A : kA = 0}. ,ym free m o d u l e Y1 is not a unit determined The f o l l o w i n g Z-torsion for where 6i~)J will be the d e s i r e d element. = ±i. 20 (See [L, Cor. =l free module of type §8. K. Z - t o r s i o n - f r e e modules If (~) is the annihilator minimal polynomial of A. ideal of A, we refer to For each prime factor consider the T - p r i m a r y s e q u e n c e s of n of l l, as a we may A: 0 + Ai+ 1 ÷ A i ÷ A i + A i+l ÷ 0 c o n s t r u c t e d in §i where derivatives of A i = 0, factors of l 7, i, n.

Then Thus ~ = ~kB' n-primary A = B E ¢(A). A-modules of the if and only if A0 ~ B0" Proof: §16. 4. 1. by induction on k. sequences: {0 + Ai+ 1 + A. ÷ A i ÷ A i+l + 0} i where = Ak+ 1 A k+l = 0, we first construct {0 ÷ Bi+ 1 ÷ B i + B i + B i+l + 0} with By exists #kB induction = 0 we may whose assume n-primary there sequences {0 ÷ Bi+ I + B I• + B i + B i+l + 0} . 1, let C be an k = i, with elementary of d e g r e e k (#C) 0 ~ C O . 3. consisting and induced map We define of all then elements C O = A k.