I'm trying to prove the GoingUp theorem from Commutative Algebra using a different method to that given in the classic reference Atiyah and Macdonald. There's a couple of parts I'm having trouble with.
All rings are commutative.
 Let A be a subring of B
 Let B be integral over A
 Let
be a prime ideal of A
1. Let
be an ideal of B which is maximal subject to
. Show that

is a maximal ideal of B

, where
(solved)
NOTE: Not sure if this should be "a maximal ideal of B", or "an ideal of B, maximal subject to
".
2. Let
be ideals of B such that
. Show that

is NOT a prime ideal of B [HINT: Consider integrality in
].
3. Suppose that
is a maximal ideal of B, such that
. Show that

. (solved)
4. Using 13, deduce that if
are prime ideals of A and
is a prime ideal of B such that
then there exists a prime ideal
of B such that


.
Any help would be greatly appreciated  I have made some progress, for example in Q1 I am fairly sure you have to suppose for a contradiction you can find an ideal $\mathfrak{r}\supsetneq\mathfrak{q}$ and then let $x\in \mathfrak{r}\setminus \mathfrak{q}$ and somehow show $(\mathfrak{q}+Bx)\cap A\subseteq \mathfrak{p}$. Not sure how to do this.
In 2 it may be helpful to know two theorems:
 If B is integral over A, q is an ideal of B, and P = A int q, then B/q is integral over A/P.
 if B is integral over A and both are integral domains, then A is a field if and only if B is a field.