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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 Variants of Rado's selection lemma and their applications, Math. Nachr. 79 (1977), 145--165 states:

Theorem 4.1. 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 $\mathfrak{p} \cap S= \emptyset$.

Rav also showed:

Corollary 4.4. The following statements are mutually equivalent in ZF set theory:
(a) Every filter on a set can be extended to an ultrafilter.
(b) In every commutative associative ring with identity, every proper ideal is included in some prime ideal.
(c) In every Boolean algebra, every proper ideal (resp. filter) is included in some prime ideal (resp. ultrafilter).
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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 Variants of Rado's selection lemma and their applications, Math. Nachr. 79 (1977), 145--165 states:

Theorem 4.1. 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 $\mathfrak{p} \cap S= \emptyset$.

Rav also showed:

Corollary 4.4. The following statements are mutually equivalent in ZF set theory:
(a) Every filter on a set can be extended to an ultrafilter.
(b) In every commutative associative ring with identity, every proper ideal is included in some prime ideal.
(c) In every Boolean algebra, every proper ideal (resp. filter) is included in some prime ideal (resp. ultrafilter).