Exercise 10, § 7

Let A be a ring. (a) Let B be an integral A-algebra. Prove that for every element a of A and every ideal a of A, the element a is integral over a if and only if a is integral over the ideal a.B generated by a in B. (b) Assuming A noetherian and reduced, denote by $p_1, ..., p_r$ the minimal prime ideals of A and consider the natural injection:

Prove that it makes $\prod_{1 \leqslant i \leqslant r} A/p_i$ an integral A-algebra. (c) Deduce that, given a noetherian ring A, for an element a of A to be integral over an ideal a, it is necessary and sufficient that, for every minimal prime ideal p of A, the image of a in A/p is integral over the ideal $(a + p)/p$. (d) Prove that if $\overline{a} = \overline{b}$, the ideals a and b have the same associated minimal prime ideals.