Exercise 8, § 1

Given a ring $A$, a polynomial $P(X_1, \ldots, X_n)$ with coefficients in $A$ is called dominated if it is of the form $X^\alpha + \sum_\beta c_\beta X^\beta$, where $\beta < \alpha$ in the product ordered set $\mathbf{N}^n$ for all $\beta$ such that $c_\beta \neq 0$. In a ring $R$, a subring $A$ is called paravaluative if, for every dominated polynomial $P \in A[X_1, \ldots, X_n]$ (any $n$), $P(s_1, \ldots, s_n) \neq 0$ for all elements $s_i \in R - A$ ($1 \leq i \leq n$).

(a) Show that, if $A$ is paravaluative in $R$, the product of two elements of $S = R - A$ belongs to $S$.

(b) Let $A$ be a subring of a ring $R$ such that the product of two elements of $S = R - A$ belongs to $S$; let $p_0$ be the prime ideal of $A$ and $R$ consisting of the $a \in A$ such that $sa \in A$ for all $s \in S$ (Exercise 6 (b)). For $A$ to be paravaluative in $R$, it is necessary and sufficient that $A/p_0$ be paravaluative in $R/p_0$.

(c) Let $R$ be a ring and $h : R \to R'$ a ring homomorphism. If $A'$ is a paravaluative subring of $R'$, $A = h^{-1}(A')$ is paravaluative in $R$. Deduce that, if $(A, p)$ is a maximal ordered pair in $R$ (Exercise 7), $A$ is paravaluative in $R$ (use (b) and Exercise 7 (d) and (e)).

(d) Let $R'$ be a ring and $R$ a subring of $R'$. If $A$ is a paravaluative subring of $R$, show that there exists a paravaluative subring $A'$ of $R'$ such that $A' \cap R = A$. (If $S = R - A$ and $S_0 = S \cup \{1\}$, consider the ring $T' = S_0^{-1}R'$ and the canonical homomorphism $h : R' \to T'$. Let $B$ be the subring of $T'$ generated by $h(A) \cup (h(S))^{-1}$ and $b$ the ideal of $B$ generated by $(h(S))^{-1}$. Show that $b \neq B$; consider a maximal ordered pair $(A', p'')$ in $T'$ such that $B \subset A''$ and $b \subset p''$ and take $A' = \overline{h}(A'')$.)

Conclude from this that, if $R$ is an integral domain, the paravaluative subrings of $R$ are the intersections of $R$ with the valuation rings of the field of fractions of $R$.