8.5. FINITELY GENERATED MODULES OVER A PID, PART II. 381 (b) The decomposition in part (a) is unique, in the following sense: Suppose M Š R=.b 1 / ˚ R=.b 2 / ˚ ±±± ˚ R=.b t / ˚ R ` ; where the b i are nonzero, nonunit elements of R , and b i divides b j for i ² j . Then s D t , ` D k and .a i / D .b i / for all i . Before addressing the uniqueness statement in the theorem, we intro-duce the idea of torsion . Suppose that R is an integral domain (not necessarily a PID) and M is an R –module. For x 2 M , recall that the annihilator of x in R is ann .x/ D f r 2 R W rx D0 g , and that ann .x/ is an ideal in R . Since 1x D x , ann .x/ ³ ¤ R . Recall from Example 8.2.6 that Rx Š R= ann .x/ as R –modules. An element x 2 M is called a torsion element if ann .x/ ¤ f0 g , that is, there exists a nonzero r 2 R such that rx D0 . If x;y 2 M are two torsion elements then sx C ty is also a torsion element for any s;t 2 R . In fact, if r 1 is a nonzero element of R such that r 1 x D0 and r 2 is a nonzero element of
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