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The operators on finite dimensional vector spaces over any field can be seen in nice form w.r.t. some choice of basis such as "diagonal, triangular, Jordan form, rational form etc." But, for operators on finitely generated modules over PID, I didn't find such theory (including minimal polynomial, characteristic polynomial, minimal polynomial divides characteristic polynomial).

Can one suggest some reference for this theory (for finitely generated modules over PID)?

I want to know about:

(i) Concepts of minimal polynomial ($m(x)$) and characteristic polynomial ($c(x)$) for operators on finitely generated modules over PID, and whether $m(x)|c(x)$,

(ii) Jordan form of an operator

(iii) Rational form of an operator

(iv) Diagonalizable and triangulable operators.

Thanks in advance.

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    @QiaochuYuan: I vaguely remember someone telling me that, albeit difficult, the classification of pairs of commuting endomorphisms existed. Does that ring a bell?2012-07-17

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The Smith normal form works over every PID.

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    Yes, but (i) you multiply by different matrices on the right and on the left, so Smith normal form is really about linear maps from a module to *another* module, not about endomorphisms; (ii) the modules have to be free.2012-07-17