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What is the (natural) bijection between the isomorphic class of sub modules and isomorphic class of quotient modules of a finitely generated torsion module over a PID. Is there any inclusion relation between these classes?

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$(N \subset M) \leftrightarrow (M / N)$ –  name Apr 12 '13 at 12:49
    
My intuitive guess is also this same correspondence. but I could not prove that non isomorphic sub modules have non isomorphic quotient modules.. is this very obvious? thanks. –  GA316 Apr 12 '13 at 13:43
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It's not true in general that non-isomorphic submodules give non-isomorphic quotients. For example, in the abelian group (i.e., $\mathbb Z$-module) $(\mathbb Z/4)\times(\mathbb Z/2)$, the cyclic subgroup $(\mathbb Z/4)\times0$ and the non-cyclic subgroup $(2\mathbb Z/4)\times(\mathbb Z/2)$ both produce quotients that are cyclic of order 2. –  Andreas Blass Apr 12 '13 at 14:04
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up vote 0 down vote accepted

If $M=R/(p^n)$ with $p$ prime, the result is clear. Since an arbitrary $M$ is a direct sum of such modules, the result is still clear.

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I try to prove the arbitrary case, for a particular sub module I could not identify the corresponding quotient module. can you explain me. –  GA316 Apr 12 '13 at 17:28
    
thanks. general case I got some idea. –  GA316 Apr 12 '13 at 17:45
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