The answer is Yes, and this is the ultraproduct construction. Let U be any nonprincipal ultrafilter on the set of primes. This is simply the dual filter to a maximal ideal on the set of primes, containing all finite sets of primes. (In other words, U contains the Frechet filter.)
The quotient R/U is a field of characteristic 0. The most important theorem is Los's theorem, which says that a first order statement about [f]_U is true in R/U just in case the set of p for which it is true in F_p about f(p) is in U. Since for most F_p, the number 1+...+1 (n times) is not 0, for n > 0, we get that this statement is true for all n in R/U.
Similarly, every F_p is a field, so this is also true in R/U.