two local rings containing the same local ring

Question

Let $f:(A,\mathfrak{m})\rightarrow(B,\mathfrak{n})$ be a local ring homomorphism such that $f$ preserves a sub-local ring, i.e. there exists a local ring $(A',\mathfrak{m}')\subsetneq(A,\mathfrak{m})$ such that $(A',\mathfrak{m}')\cong f(A')\subset (B,\mathfrak{n})$ as a local ring. In this case, are our two local rings $(A,\mathfrak{m})$ and $(B,\mathfrak{n})$ isomorphic? If not, is there any counterexamples?

Answer 1

Take three fields $L/K/E$ such that all of these extensions are proper. Then let $f: K \longrightarrow L$ be the inclusion. Any map between fields is local (here I assume ring homomorphisms are unital). Furthermore, $f|_E$ maps $E$ isomorphically onto $f[E]$. Of course, $f|_E$ is the identity on $E \longrightarrow f[E]$.

EDIT: Sorry, I clicked submit too early. I should also add the stipulation that these three fields are not isomorphic. Concretely, take $\mathbb C / \mathbb R / \mathbb Q$. If you don't like that my examples were fields, take $L/K/\mathbb Q$ distinct number fields. Then let $p \in \mathbb Z$ prime and $\mathfrak p \subseteq \mathcal O_K$ a prime over $(p)$ and $\mathfrak q \subseteq \mathcal O_L$ a prime over $\mathfrak p$. Then consider the extensions of local rings $\mathbb Z_{(p)} \subseteq (\mathcal O_K)_{\mathfrak p} \subseteq (\mathcal O_L)_{\mathfrak q}$. Take again $f$ to be the inclusion $(\mathcal O_K)_{\mathfrak p} \longrightarrow (\mathcal O_L)_{\mathfrak q}$. It maps $\mathbb Z_{(p)}$ isomorphically onto itself, but $(\mathcal O_K)_{\mathfrak p} \not \cong (\mathcal O_L)_{\mathfrak q}$.