In the language of modules, it suffices to restrict our view to *positive-primitive* formulas - that is to say, formulas with one existential quantifier and no negation.


And I mean *existentially closed* in the sense that any witness to a particular positive-primitive formula over the module is already in the module itself.


My intuition is that I should look along the lines of injective modules. Can anyone help by offering direction or perhaps an example / verification?



Example or classification of existentially closed modules
Do primitive positive formulas only allow conjunctions, not disjunctions? If so, then injective modules are existentially closed, but are a strictly stronger concept in that the equivalent for infinite conjunctions are allowed.


The concept you're after is that of [algebraically compact](http://en.wikipedia.org/wiki/Algebraically_compact_module) module (also known as pure inective module). Here's a link to a paper by [Prest](http://eprints.ma.man.ac.uk/1148/01/covered/MIMS_ep2008_83.pdf) that talks about some their properties.


[Baer's criterion](http://en.wikipedia.org/wiki/Injective_module#Baer.27s_criterion) asserts existential closure under infinite conjunctions. Let
$$
(\exists m) r\_1 m = m\_1 \wedge r\_2 m = m\_2 \wedge \cdots
$$
be a particular formula, where $R$ is your coefficient ring, $M$ a module over $R$, with $r\_i \in R$ and $m\_i \in M$. Then consider the ideal $I$ generated by $r\_i$ in $R$. There the map that sends $r\_i$ to $m\_i$ is a homomorphism from $I$ to $M$ if and only if that formula is consistent. (There's a homomorphism if and only if the formula does not imply $s = s'$ for two distinct elements $s, s' \in R$.)


