Lemma 10.16.3. Let R be a ring. Let I \subset R be an ideal. Let M be a finite R-module. Let \varphi : M \to M be an endomorphism such that \varphi (M) \subset IM. Then there exists a monic polynomial P = t^ n + a_1 t^{n - 1} + \ldots + a_ n \in R[T] such that a_ j \in I^ j and P(\varphi ) = 0 as an endomorphism of M.
Proof. Choose a surjective R-module map R^{\oplus n} \to M, given by (a_1, \ldots , a_ n) \mapsto \sum a_ ix_ i for some generators x_ i \in M. Choose (a_{i1}, \ldots , a_{in}) \in I^{\oplus n} such that \varphi (x_ i) = \sum a_{ij} x_ j. In other words the diagram
\xymatrix{ R^{\oplus n} \ar[d]_ A \ar[r] & M \ar[d]^\varphi \\ I^{\oplus n} \ar[r] & M }
is commutative where A = (a_{ij}). By Lemma 10.16.1 the polynomial P(t) = \det (t\text{id}_{n \times n} - A) has all the desired properties. \square
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