Y. Rav proved this using the Ultrafilter Principle ("Every filter on a set can be extended to an ultrafilter"), which is weaker than the Axiom of Choice. Theorem 4.1 of <a href="http://www.ams.org/mathscinet-getitem?mr=476530">Variants of Rado's selection lemma and their applications, Math. Nachr. 79 (1977), 145--165</a> states:<blockquote><b>Theorem 4.1.</b> Let R be a ring, $\mathfrak{a}$ a proper ideal in R, and suppose that S is multiplicative subsemigroup of R which does not meet $\mathfrak{a}$. Then it follows from the Ultrafilter Principle that their exists a prime ideal $\mathfrak{p}$ in R such $\mathfrak{a} \subseteq \mathfrak{p}$ and $\mathfrak{p} \cap S= \emptyset$.
</blockquote>

Rav also showed:
<blockquote>
<b>Corollary 4.4.</b> The following statements are mutually equivalent in ZF set
theory:<br />

(a) Every filter on a set can be extended to an ultrafilter.<br />

(b) In every commutative associative ring with identity, every proper ideal is included
in some prime ideal.<br />

(c) In every Boolean algebra, every proper ideal (resp. filter) is included in some
prime ideal (resp. ultrafilter).
</blockquote>