Exercise 12, § 1

(a) Let A be a ring, I and J two ideals of A. Suppose that for every maximal ideal m of A, we have IA_m = JA_m. Deduce that we have I = J. (b) Let A be a ring such that, for every maximal ideal m of A, the ring A_m is noetherian and such that, for every nonzero element f of A, there exists only a finite number of maximal ideals of A containing f. Then A is noetherian. (Let a be an ideal of A. We shall first show that there exists a finite family (x_1, ..., x_n) of elements of a such that the maximal ideals of A containing {x_1, ..., x_n} contain a. We shall then construct a finite family of elements y_1, ..., y_l of a such that for every maximal ideal m of A, {y_1, ..., y_l} generates the ideal aA_m of A_m. We shall conclude by using a). (c) Let A be a semi-local ring such that, for every maximal ideal m, the ring A_m is noetherian. Then A is noetherian.