# What is the difference between First-Order Structures and Kripke Structures?

In the SEP article on Model Theory by Wilfrid Hodges (here), he writes:

Particular kinds of model theory use particular kinds of structure; for example mathematical model theory tends to use so-called first-order structures, model theory of modal logics uses Kripke structures, and so on.

I take it that by a "Kripke structure" here, he simply means a Kripke model--- taken to be an ordered quadruple $\langle D, W, R, I \rangle$ where $D$ is a non-empty set of elements (the "domain of discourse"), $W$ is the set of "worlds", $R$ is the "accessibility relation", and $I$ is the interpretation function which assigns to every constant a member of $D$ and assigns to each world-predicate pair $\langle w, P\rangle$ an ordered n-tuple of $D$ (where n is determined by the arity of $P$).

I also take it that by a "first-order structure" he simply means the usual notion of a model in first-order model theory--- taken to be an ordered pair $\langle D, I \rangle$ where $D$ is a non-empty set of elements (the "domain of discourse") and $I$ is an interpretation function which assigns to every constant in the language an element of $D$, assigns to every n-ary predicate an ordered n-tuple of elements of $D$, and so on.

Is the only difference between these two sorts of structures the presence of an accessibility relation and a set of worlds in Kripke structures?

Could you do "mathematical model theory" with Kripke structures instead of first-order structures (that is, could it been done in any interesting sense, not simply by having a set of worlds and an accessibility relation that plays no role in the theory) ?