It is well-known that an abelian group is free iff it is projective, which can be easily extended to modules over PID. The standard proof requires the use of Zorn's Lemma/Well-ordering principle. Are there ways of proving this without axiom of choice or anything equivalent? Any hints or proof outlines will be appreciated. Any tools from introductory homological algebra and commutative algebra can be freely used. Thanks in advance.
2026-04-24 18:55:34.1777056934
Abelian group is free iff projective
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Andreas Blass proved in
Blass, Andreas, Injectivity, projectivity, and the axiom of choice, Trans. Am. Math. Soc. 255, 31-59 (1979). ZBL0426.03053.
that the statement "every free abelian group is projective" is equivalent to the axiom of choice.