Exercise 13, § 3

Let $A$ be a Noetherian local ring with maximal ideal $m$ and $B$ a ring such that $A \subset B \subset \hat{A}$. Suppose that $B$ is a Noetherian local ring with maximal ideal $n$ (Exercise 14).

(a) Show that $n = B \cap m$ and $B = A + n$. If further $n^2 = B \cap m^2$, then $n = Bm$ (note that then $n = Bm + n^2$).

(b) Let $K$ be a commutative field, $C$ the polynomial ring $K[X]$, $p$ the prime ideal $CX$, $A$ the (Noetherian) local ring $C_p$ and $m$ its maximal ideal; the completion $\hat{A}$ is identified with the ring of formal power series $K[[X]]$ (no. 4, Proposition 8). Let $u = u(X)$ be an element of $\hat{A}$ which is transcendental over the field of rational fractions $K(X)$ (cf. Algebra, Chapter V, §5, Exercise 13 or Functions of a Real Variable, Chapter III, § 1, Exercise 14 (a)) with no constant term; let $B$ be the subring of $\hat{A}$ consisting of the quotients

where $P$ and $Q$ are polynomials in $K[X, Y]$ such that $Q(0, 0) \neq 0$. Show that $B$ is a Noetherian local ring containing $A$, whose maximal ideal $n$ is generated by $X$ and $u(X)$; but on $B$ the $n$-adic topology is strictly coarser than the $m$-adic topology.

(c) With the same definitions as in (b), let $B'$ be the subring of $B$ generated by $A$ and $u(X)$; show that $B'$ is not a local ring (note that $B' \cap n$ is a maximal ideal of $B'$ but that there are elements of $B'$ not invertible in $B'$ and not belonging to $B' \cap n$).