Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 4 / 11G / a

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Solution

The structure theorem for finitely generated modules over a principal ideal domain says that a finitely generated module over a Euclidean domain

R

is

R^{r} \oplus R / (d_{1}) \oplus \dots \oplus R / (d_{k}), d_{1} ∣ d_{2} ∣ \dots ∣ d_{k},

with the rank and nonunit invariant factors of a linear operator unique up to associates.

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