Let A be a ring which is commutative and unitary. An ideal I of A is said to be almost maximal iff it is prime and if it is equal to its Jacobson ideal. Using topological products and transfinite recursion, we show that the two following axioms are equivalent:
BPI: Every non-trivial boolean algebra has a prime ideal.
AMIT: Every non-trivial commutative unitary ring has an almost maximal ideal.

This solves a question raised by Johnstone, [J84] . This question has also been solved by Blass, see [Bl87], using logical tools, and by Banachewski, see [Ba85], using algebraic tools .