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cyclic module

SimonLee
in Linear Algebra and Group Theory

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Suppose R is a PID which is not field, M is a finitely generated module on R. prove that if for any prime p, M/pM is cyclic R/pR module, then M is cyclic.

 

I am just trying using the uniqueness of the structure theorem about finitely generated module over pid...but don't know how to connect M with M/pM. Any help would be appreciated.

 

Simon

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