8.5. FINITELY GENERATED MODULES OVER A PID, PART II. 385 as an R=.p/ –vector space. Applying the same considerations to the other direct sum decomposition, we obtain that the number of b i divisible by p is also equal to dim R=.p/ .M=pM/ . If p is an irreducible dividing a 1 , then p divides all of the a i and exactly s of the b i . Hence s ± t . Reversing the role of the two decom-positions, we get t ± s . Thus the number of direct summands in the two decompositions is the same. Fix an irreducible p dividing a 1 . Then p divides a j and b j for 1 ± j ± s . Let k0 be the last index such that a k0 =p is a unit. Then pA j is cyclic of period a j =p for j > k0 , while pA j D f0 g for j ± k0 , and pM D pA k0 C 1 ˚ ²²² ˚ pA s . Likewise, let k 00 be the last index such that b k 00 =p is a unit. Then pB j is cyclic of period b j =p for j > k 00 , while pB j D f0 g for j ± k 00 , and pM D pB k 00 C 1 ˚ ²²² ˚ pB s . Applying the induction hypothesis to
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