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Let $F$ be any non-principal ultrafilter on $\mathbb{N}$. Let $I$ be the subset of the ring $R:=\prod_1^{\infty}\mathbb{Z}$ consisting of all functions that vanish on some element of $F$. Then $I$ is an ideal containing $J:=\coprod_1^{\infty} \mathbb{Z}$, whence we get a surjective homomorphism $R/J\to R/I$. But $R/I$ is isomorphic to the ultrapower of the integers determined by $F$. By Łoś's theorem, $R/I$ is an elementary extension of $\mathbb{Z}$. It follows that the quotient $(R/I)/p(R/I)$ is isomorphic to $\mathbb{Z}/p \mathbb{Z}$. For terminology and details, see The Use of Ultraproducts in Commutative Algebra, by Hans Schoutens.