Minimal primes and ring extensions?

I think yes

First, let M=R-p

Since p is prime, M is a multiplicative set in S that does not contain 0.

Let N={I ideal in S| I intersected M=empty set}

N is nonempty (it contains 0} and satisfies the condition of Zorn's Lemma

So it has a maximal element q; which can be proved is prime

(take ab in q, a,b not in Q; so (q+Sa) intersected M and

(q+Sb) intersected M nonempty...

(this appears also in Matsumara, Commutative algebra, page 5)

Now q intersected R is a prime ideal in R that by construction of q is included in p.

Since p is minimal, the intersection is p.

no comments?!

Have you tested this on the ring of the integers?