Say $R$ is a ring, and $I_1,...,I_n,J\subseteq R$ ideals, s.t $J\subseteq\bigcup_{i=1}^nI_i$. If there exists $\phi:K\rightarrow R$ homomorphism where $K$ is an infinite field, then there exists an $i$ s.t $J\subseteq I_i$.
I split it into two cases: Since $\phi$ is an homomorphism and $K$ is a field, then $\phi^{-1}(J)=\langle0 \rangle $ or $\phi^{-1}(J)=K$. If it's the first case, then since $\phi$ is injective (since $K$ is a field) we then get that $J=\langle 0 \rangle$ and obviously for every $i$, $J\subseteq I_i$.
Now, if $\phi^{-1}(J)=K$, then there must be at least one index $i$ s.t $\phi^{-1}(I_i)=K$ - otherwise the union is $\bigcup_{i=1}^n I_i=\langle 0 \rangle$ - contradiction since $J\neq\langle 0 \rangle$. So I can see that $\phi(\phi^{-1}(J))\subseteq \phi(\phi^{-1}(I_i))$, but this doesn't immediately mean that $J\subseteq I_i$ - even with $\phi$ being an injection.
What am I missing here? Any help would be appreciated.