Exercise 6, § 1

Let R be a ring and A a subring of R such that S = R − A is non-empty and the product of two elements of S belongs to S.

(a) Let $a, a'$ be two elements of A and s, s' two elements of S such that $sa \in A$ and $s'a' \in A$. Show that one of the two elements $sa', s'a$ belongs to A. Deduce that the set $p_1$ of elements $a \in A$ for which there exists $s \in S$ such that $sa \in A$ is a prime ideal of A.

(b) Show that the set $p_0$ of $a \in A$ such that $sa \in A$ for all $s \in S$ is a prime ideal of R and A; it is the greatest ideal of R contained in A and $p_0 \subset p_1$.

(c) The set n of elements $x \in R$ such that there exists $s \in S$ for which $sx = 0$ is both an ideal of R and an ideal of A and $n \subset p_0$.

(d) The ring A is integrally closed in R.

(e) If R is a field, A is a valuation ring of R, $p_0 = (0)$ and $p_1$ is the maximal ideal of A.