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finitely generated modules over a principal ideal domain

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The statement could be made more precise in the following manner:

If M is a finitely generated module over a PID R, then

M \cong R^n \oplus R/(q_1) \oplus ... \oplus R/(q_r)

where q_1 | q_1 | ... q_r, the ideals (q_i) are uniquely determined.

Here, R^n is the free module F which is the torsion free part of M. Therefore, n is uniquely determined too. The other factors represent a direct sum decomposition of M_{tor} (the torsion submodule of M).

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