Injecttvtty , Projecttvtty , and the Axiom of Choice

We study the connection between the axiom of choice and the principles of existence of enough projective and injective abehan groups. We also introduce a weak choice principle that says, roughly, that the axiom of choice is violated in only a set of different ways. This principle holds in all ordinary Fraenkel-Mostowski-Specker and Cohen models where choice fails, and it implies, among other things, that there are enough injective abehan groups. However, we construct an inner model of an Easton extension with no nontrivial injective abehan groups. In the presence of our weak choice principle, the existence of enough projective sets is as strong as the full axiom of choice, and the existence of enough free projective abehan groups is nearly as strong. We also prove that the axiom of choice is equivalent to "all free abehan groups are projective" and to "all divisible abelian groups are injective." The classical development of homological algebra [4] depends upon two theorems asserting that every module, over an arbitrary ring, is both a homomorphic image of a projective module and a submodule of an injective module. The usual proofs of these two theorems make use of the axiom of choice. Our purpose here is to investigate the connection between the axiom of choice and the principles of existence of enough projectives and enough injectives. In very rough terms, our results can be summarized by saying that the existence of enough injectives is a very weak form of choice, while the existence of enough projectives is rather strong. In §1, we establish our notation and some preliminary results, for example that it suffices to consider abelian groups instead of modules over an arbitrary ring. §2 is devoted to the proof that the axiom of choice is equivalent to key ingredients of the usual proofs of the theorems that there are enough projective and enough injective abehan groups. Thus, any proofs of these two theorems using less than the full strength of the axiom of choice would have to proceed along different lines. In §3, we construct a model of set theory with no nontrivial injective abelian groups. It is a permutation model in which the atoms (= urelements) form a proper class; the same result can be obtained without atoms by means of forcing with a proper class of conditions. The need for proper classes in the construction of §3 is shown in the next two sections. In §4, we isolate a property that distinguishes models Received by the editors March 6, 1978. AMS (MOS) subject classifications (1970). Primary 02K20 (= 04A25); Secondary 02K05, 02K15, 13C10, 18G05. © 1979 American Mathematical Society 0002-9947/79/0000-0501 /$08.2S 31 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use