Exercise 18, § 1

Let A be an integral domain, K its field of fractions, M a finitely generated A-module, u an endomorphism of M and u @ 1 the corresponding endomorphism of the finite dimensional vector K-space M_{(K)} = M ⊗_A K. Show that the coefficientsof the characteristic polynomial of u ⊗ 1 are integral over A (use condition (E_{III}) of Theorem 1 of no. 1).