Exercise 5, § 4

Let A be a noetherian ring satisfying conditions (R1) and (S2). a) Show that A is reduced. b) Let $a$ be a cancellable element of A. If an element $x$ of A is such that its image in $A_p$ belongs to $aA_p$ for every $p \in \mathrm{Ass}_A(A/aA)$, then $x$ belongs to $aA$. (Use IV, § 2, No. 3, Prop. 5.) c) Prove that A is integrally closed in its total ring of fractions R. (If $a, b, c_1, ..., c_n$ are elements of A satisfying (i) $b$ is cancellable in A, (ii) $(b^{-1}a)^n + c_1(b^{-1}a)^{n-1} + \cdots + c_n = 0$ in R, then prove successively that one has $a \in bA_p$ for every prime ideal $p$ of height 1 of A, and then that one has $a \in bA$.) d) Prove that A is normal. (Remark that an idempotent of R is integral over A.)