Kaplansky in his paper titled by Projective Modules gave an important and essential theorem as follow:
Theorem: Let $R$ be a ring, $M$ an $R$-module which is a direct sum of (any number of) countably generated $R$-modules. Then any direct summand of $M$ is likewise a direct sum of countably generated $R$-modules.
But if you could take a look to the pattern of his proof, he applied the well ordering Theorem for proving it.
I am thinking about the relation of his proof with the well ordering theorem. More precisely I am thinking about the answer of the following question:
Question: Is the Kaplansky theorem equivalent with the Axiom of Choice or it can be proved with the weeker axiom(i.e.boolean prime ideal theorem)?