I was trying to solve a problem involving local rings and I did the following, which seems to lead to a contradiction but I cannot find where I have messed up:
For a field $\mathbb{K}$ we know that $\mathcal{M}=(x)/(x^3)$ is the unique maximal ideal in the quotient ring $\mathbb{K}[x]/(x^3)$. Let $q: \mathbb{K}[x] \rightarrow \mathbb{K}[x]/(x^3)$ be the quotient map, $\mathcal{I} = (x)$ and $\mathcal{J} = (x^3,1+x)$. Then $q(\mathcal{J}) \subseteq q(\mathcal{I})$ since $q(\mathcal{I}) =\mathcal{M}$ is a unique maximal ideal. Finally this means that $q^{-1}(q(\mathcal{J}))\subseteq q^{-1}(q(\mathcal{I})) \implies \mathcal{J} \subseteq \mathcal{I}$, which is false.
Where have I gone wrong?
$q(\mathcal{J})$ is not contained in any maximal ideal. You may be confused because every proper ideal is contained in a maximal ideal. The point of course is that $q(\mathcal{J})$ is not a proper ideal.