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?