Exercise 8, § 1

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

a) Now prove 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 $\mathfrak{p}_0$ be the prime ideal of $A$ and of $R$ formed by the $a \in A$ such that $sa \in A$ for every $s \in S$ (exerc. 6 b)). In order that $A$ be paravaluative in $R$, it is necessary and sufficient that $A/\mathfrak{p}_0$ be paravaluative in $R/\mathfrak{p}_0$.

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

d) Let $R'$ be a ring, $R$ a subring of $R'$. If $A$ is a paravaluative subring of $R$, now prove that there exists a paravaluative subring $A'$ in $R'$ 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 $\mathfrak{b}$ the ideal of $B$ generated by $(h(S))^{-1}$. Prove that $\mathfrak{b} \neq B$; consider a maximal pair $(A'', \mathfrak{p}'')$ for $T'$ such that $B \subset A''$ and $\mathfrak{b} \subset \mathfrak{p}''$, and take $A' = h^{-1}(A'')$).

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