The class of models of the positive theory of fields

Question

Let $T$ be the deductive closure of the theory of fields in the signature $(+,-,*,0,1)$, and let $T'$ be the deductive closure of the positive formulas of $T$. Is it true that $Mod(T)$ is the class of fields united with the class of singleton rings? What, precisely, is the class of models of the positive theory of fields? Just to clarify, a positive formula is one built up from atomic formulas using disjunction, conjunction, universal quantifiers, and existential quantifiers.

Answer 1

Lyndon's preservation theorem says that the positive formulas in your sense are exactly those preserved under quotients, aka surjective homomorphisms, aka homomorphic images. As a consequence, an elementary class of structures is closed under quotients if and only if it is axiomatized by a positive theory (see p. 67 of these notes).

So the positive theory of fields has as its class of models the smallest elementary class containing the fields and closed under quotients. The image of a field $K$ under a ring homomorphism is still a ring, and if it is not isomorphic to $K$, then the kernel of the homomorphism must be the only other ideal in $K$, namely the unit ideal. So the class of quotients of fields is exactly the fields together with the trivial ring. This is already an elementary class, so it's $\text{Mod}(T')$, as you guessed.