Question

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?

Answer 1

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 module (also known as pure inective module). Here's a link to a paper by Prest that talks about some their properties.

Baer's 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 $s, s' \in R$.)