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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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This is equivalent to the study of modules over $R[x]$ (where $R$ is your PID) which seems difficult. For $R = k[y]$ this is the study of modules over $k[x, y]$, or the study of pairs of commuting finite-dimensional linear operators up to conjugation. –  Qiaochu Yuan Jul 17 '12 at 3:08
    
@QiaochuYuan: I vaguely remember someone telling me that, albeit difficult, the classification of pairs of commuting endomorphisms existed. Does that ring a bell? –  PseudoNeo Jul 17 '12 at 20:34
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1 Answer 1

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. –  PseudoNeo Jul 17 '12 at 7:55
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