Exercise 8, § 1

Let A be a commutative ring, $B = A[X_n]_{n \geq 1}$ a polynomial ring over A in an infinity of indeterminates and C the quotient ring of B by the ideal generated by the polynomials $(X_1 - X_2)X_i$ for $i \geq 3$. If $\xi_1$ and $\xi_2$ are the classes in C of $X_1$ and $X_2$ respectively, show that the intersection in C of the monogenous ideals $C\xi_1$ and $C\xi_2$ is not a finitely generated ideal.*