When studying David Marker's "Model Theory: An Introduction" book trying to understand the proof of Lemma 3.2.1 which says: $ACF_{\forall}$ is the theory of integral domains, I couldn't understand the last line of the proof which comes as followa: (I copy the whole proof)
Proof. The axioms for integral domains are universal consequences of $ACF$. If $D$ is an integral domain, then the algebraic closure of the fraction field of $D$ is a model of $ACF$. Because every integral domain is a subring of an algebraically closed field, $ACF_{\forall}$ is the theory of integral domains.
David Marker wants to show $ACF_{\forall}$ is the theory of integral domains, he is just satisfied with proving, as in another book :Model theory notes by Kevin Buzzard(which you can find it in page 12-13 from the link below*), that these two theories have the same models. But I myself think we have to show $ACF_{\forall} = ID$. The author has already proved that $Mod(ACF_{\forall}) = Mod(ID)$ and it is easy to show $ID\subseteq ACF_{\forall}$ also ,but I can't show the converse inclusion. Do you think if $Mod(T_{1}) = Mod(T_{2})$ and $T_{1}\subseteq T_{2}$, then we have $T_{2}\subseteq T_{1}$ for two theories $T_{1}$ and $T_{2}$? If yes our problem is solved? or ...
I also asked this question in this manner which couldn't lead me to my very mean if you want to see:
"A question about two theories and their models "
Please say if my understanding of the original proof is true or not? and help to prove it, please.
*http://wwwf.imperial.ac.uk/~buzzard/maths/research/notes/model_theory_notes.pdf
The use of the phrase "is the theory of integral domains" in the statement and proof of Marker's Lemma 3.2.1 is very sloppy. What Marker actually means is that $ACF_{\forall}$ axiomatizes the class of integral domains and that is what the proof proves (and is what is needed at the point of use of the lemma to show that $ACF$ has algebraically prime models).
It is clearly wrong to say that $ACF_{\forall}$ is the (full) theory of integral domains (which is the only notion of theory of a class of structures that Marker has defined), as there are sentences like $\forall x \exists y\cdot x + y = 0$ that hold in any integral domain but are not purely universal.
The closest you can get to a true statement along the lines of Marker's actual statement of the lemma is that $ACF_{\forall} = ID_{\forall}$, but that isn't exactly what is wanted where the lemma is used.