My main corollary is:
For each non-principal ultrafilter F on the set of primes:
$\prod (\mathbb Z/(p^n))\cong \prod (\mathbb F_p[T]/(T^n)) $
I know that the above is correct because due to the baby ax-kochen principle:
the $(\mathbb Z/(p^n))$ and $(\mathbb F_p[T]/(T^n))$ are elementarily equivalent
And by the kiesler and shellah theorem:
any two L-structure $M$ and $N$ are elementarily equivalent if there exists an index $st I$ and and ultrafilter $D$ such that $\prod M/D \cong \prod N/D$
My question is how do I define the ultrafilter on primes?
Should I define it on prime ideal or prime numbers?