how do we prove that ring of characteristic $p$ has arbitrarily large models?

Question

As title says, how do we prove that the theory that describes ring of characteristic $p$ has arbitrarily large model?

I am asking for a model-theoretic approach.

Answer 1

Add to our language a set $C$ of constant symbols of cardinality $\kappa$. Add to the usual axioms for the theory of (say) fields of characteristic $p$ the sentences $\lnot(a=b)$, for all pairs $\{a,b\}$ of distinct constant symbols in $C$.

Call the resulting theory $T$. We show that every finite subset $T^\ast$ of $T$ has a model.

The set $T^\ast$ of axioms can mention only finitely many of the constant symbols in $C$, say $n$ of them.

The theory of fields of characteristic $p$ has arbitrarily large finite models. So in particular it has a model $M$ with at least $n$ elements. This $M$ can be made into a model of $T^\ast$ by interpreting the constant symbols mentioned in $T^\ast$ as different elements of $M$.

Since every finite subset of $T$ has a model, the theory $T$ has a model, by the Compactness Theorem. But the special axioms of $T$ ensure that any such model has cardinality at least equal to the cardinality of $C$.

Remark: By refining the argument, one can show that there is in fact a model of every infinite cardinality. All that we have shown above is that for any cardinal $\kappa$, there is a model of cardinality $\ge \kappa$.

Answer 2

A different approach would be to use Skolem's theorem. For any infinite model $M$, there is an elementary submodel (or an elementary supermodel) of any cardinality smaller (resp. larger) than that of $M$, so it is enough to show that there is any infinite ring of characteristic $p$ (because being a ring of characteristic $p$ is an elementary property).

But it's easy to find such a ring, for example the ring ${\bf Z}_p[X]$ of polynomials over ${\bf Z}_p={\bf Z}/p{\bf Z}$ ($p$ need not be prime here).

Still, it strikes me as a bit of an overkill to use model theory at this point: we could have just as well pointed out that the ring ${\bf Z}_p[X_\alpha]_{\alpha<\kappa}$ of polynomials of $\kappa$ variables over ${\bf Z}_p$ has cardinality $\kappa$ for infinite $\kappa$, and incidentally, I'm pretty sure it is also an elementary chain.