Suppose I have two local rings $A$ and $B$, and suppose I have $\phi : A \rightarrow B$, which is a ring isomorphism. Does it follow then that $\phi$ is a local ring homomorphism?
The point of question is that I have two local rings, which I am trying to show that they are "the same". And I was wondering if showing that they are isomorphic is enough or not. Thanks!
If $(A, m)$ and $(B,n)$ are local rings and $\phi : A \to B$ is an isomorphism, then for any $x \in A$, $x \not \in m \iff x$ is a unit in $A \iff \phi(x)$ is a unit in $B \iff \phi(x) \not \in n$, so $\phi(m) \subseteq n$, i.e. $\phi$ is a local homomorphism.