Exercise 2, § 5

Let $K$ be a field and $B$ the ring of polynomials with coefficients in $K$ in an infinite system of indeterminates $X_i \ (i \in \mathbf{N})$; let $b$ be the ideal of $B$ generated by the products $X_i X_j$ ($i \neq j$), $A$ the quotient ring $B/b$, $\xi_i$ the class of $X_i$ mod. $b$ ($i \in \mathbf{N}$), $p$ the ideal of $A$ generated by the $\xi_i$ of index $i \geq 1$, which is prime, and $u$ the canonical mapping $A \to A/p$. Show that $u_p : A \to (A/p)_p$ is bijective but that there exists no $f \in A - p$ such that $u_f$ is bijective.