Re-interpreting vector spaces in a choice-less model of ZF as modules over a regular ring in ZFC

Question

I am searching a module $M$ over a (von Neumann) regular ring $A$ ($\forall a\in A$, $\exists x\in A$: $axa=a$) with two properties:

(1) every finitely generated submodule of $M$ is projective (isomorphic to a direct summand of some $A^n$);

(2) at least one finitely generated [projective] submodule of $M$ is not a direct summand of $M$.

[Equivalent re-formulation: a Mittag-Leffler, not strictly Mittag-Leffler module $M$ over a regular ring $A$. Another re-formulation: a sectionally complemented modular lattice $L=L(A^4\oplus M)$, with a large quadruplicable element,such that not every element of $L$ is complemented in the ideal completion. These re-formulations are not needed in what follows]

It is well known that models of ZF (without choice) exist where even vector spaces give such examples: real numbers as vector space over the rational numbers (in models of ZF where all additive endomorphisms of the reals are linear); or: amorphous sets with a vector space structure over a finite field. These examples even have a property stronger than (2):

(2') there is no nonzero linear functional on $M$ (hence no nonzero finitely generated [projective] submodule is a direct summand)

Now it comes the tricky part. Suppose that, instead of needing models that negate AC, I were interested in some easier models, such that a Boolean valued universe would be sufficient (without the need to consider a symmetric sub-model). Then vector spaces in the ZFC models obtained form a Boolean valued universe are easily re-interpretable as coming from modules over [strongly] regular rings (whose Boolean algebra of [central] idempotents is the complete Boolean algebra of truth values for the Boolean valued universe).

The questions, returning to models of ZF without choice:

Is there an analogous reinterpretation (existing in every model of ZFC) for the case I am interested in (vector spaces in models of ZF without choice)?

If such a reinterpretation is possible, what should I study to understand it? [Note that I am not even able to have a intuitive mental picture of the topos of $G$-sets for a group $G$]

If such a reinterpretation is possible, is the syntactical form of the properties (1), (2), (2') such that their validity in the ZF model automatically implies validity in the reinterpretation?

Answer 1

I cannot give you any specific example, but at least over nontrivial countable regular rings such modules always exist.

We work over a countable regular ring $A$ which is not semisimple Artinian, and our modules are, for instance, right $A$-modules. In this case, $A$ cannot be injective as a module over itself.

The first ingredient is Lemma 9.10 from [1] which says that, if you take any module $F$ such that there exists a short exact sequence $0 \to A \to M \to F \to 0$ that does not split, then $M$ is not strict Mittag-Leffler. Since $A$ is not injective, we can always find such a module $F$.

The very same lemma also ensures that $M$ is Mittag-Leffler whenever $F$ is such. The question is: can we find $F$ which is Mittag-Leffler and at the same time $\mbox{Ext}^1(F,A)\neq 0$?

This is not easy since such $F$ cannot be countably presented: indeed, every countably presented (flat) Mittag-Leffler module is projective, hence strict Mittag-Leffler.

To find the right $F$ (or to be more precise, to prove its existence), you can use Theorem 6, in particular the moreover-clause statement (1), from [2]. In the setting of countable regular rings, it says that any module $N$ ($A$ in particular) which satisfies that $\mbox{Ext}^1(G,N) = 0$, for all Mittag-Leffler modules $G$, has to be injective.

Final remark: It is an open problem whether the statement (1) in Theorem 6 holds for uncountable rings. On the other hand, in your case, the assumption that $A$ is countable can be replaced by `there exists a countably presented module $C$ such that $\mbox{Ext}^1(C,A) \neq 0$'.

[1] L. Angeleri Hügel, D. Herbera: Mittag-Leffler conditions on modules, Indiana Univ. Math. J. 57 (2008), 2459-2518.

[2] S. Bazzoni, J. Šťovíček: Flat Mittag-Leffler modules over countable rings, Proc. Amer. Math. Soc. 140 (2012), 1527-1533.