Exercise 1, § 1

Let $K$ be a field; for every subring $A$ of $K$ let $L(A)$ denote the set of local rings $A$, of $A$ where $p$ runs through the prime ideals of $A$ (these local rings being identified with subrings of $K$).

(a) For a local subring $M$ of $K$ to dominate a ring $A, \in L(A)$, it is necessary and sufficient that $A \subset M$; the local ring $A$, dominated by $M$ is then unique and corresponds to the prime ideal $p = m(M) \cap A$.

(b) Let $M, N$ be two local subrings of $K$ and $P$ the subring of $K$ generated by $M \cup N$. Show that the following conditions are equivalent:

$(a)$ There exists a prime ideal $p$ of $P$ such that $m(M) = p \cap M, m(N) = p \cap N$.

$(\beta)$ The ideal $a$ generated in $P$ by $m(M) \cup m(N)$ is distinct from $P$.

$(\gamma)$ There exists a local subring $Q$ of $K$ dominating both $M$ and $N$.

If these conditions are satisfied, $M$ and $N$ are called related.

(c) Let $A, B$ be two subrings of $K$ and $C$ the subring of $K$ generated by $A \cup B$. Show that the following conditions are equivalent:

$(a)$ For every local ring $Q \subset K$ containing $A$ and $B$, $A_1 = B_1$, where $p = m(Q) \cap A, q = m(Q) \cap B$.

$(\beta)$ For every prime ideal $r$ of $C$, $A_r = B_r$, where $p = r \cap A, q = r \cap B$.

$(\gamma)$ If $M \in L(A)$ and $N \in L(B)$ are related, they are identical.

$(\delta)$ $L(A) \cap L(B) = L(C)$.

(Use (a) and (b)).