With AC, it is easy to see that any vector space is injective, and free, therefore alse flat and projective.
Without AC, vector spaces can be not free. Are they must be projective modules? Flat modules? What about injectiveness?



Properties of vector spaces without AC
(*I completely revamped my answer, the previous version ([link](https://mathoverflow.net/revisions/163925/2)) had a consistency result with a particular example, this feels much better as it establishes a full equivalence result instead.*)


The answer is no. For projectivity and injectivity. The reason is that the axiom of choice is equivalent to the following statement:



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> If $V$ is a vector space, and $K$ is a subspace of $V$, then there exists a a direct complement for $K$, that is a subspace $S$ such that $S\cap K=\{0\}$ and $S\cup K$ is a generating set for $V$.
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If every vector space is projective, consider the case that $V$ is a vector space, $K$ is a subspace and $T\colon V\to V/K$ is the quotient map. Then there is some $h\colon V/K\to V$ such that $h\circ T$ is the identity function. In particular, this means that $K+\operatorname{im}(h)=V$, but $h$ cannot map any nonzero vector into $K$ so this is a direct sum indeed.


Therefore the statement "Every vector space is projective" implies the axiom of choice.


Similarly for injectivity, take the identity function $K\to V$, then there is a projection map whose kernel is a direct complement for $K$.




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I think that a relevant paper to mention here would be the following:



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> *Andreas Blass*, [**Injectivity, projectivity, and the axiom of choice**](http://dx.doi.org/10.2307/1998165), *Trans. Amer. Math. Soc.* **255** (1979), 31--59.
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