Could someone give an example for a homomorphism between local rings which is not local?
I tried finding the example by defining a homomorphism from Z/9Z to Z/3Z which maps a+9Z to a+3Z. But it turns out that 0,3,6(i.e. non units of Z/9Z) are mapped to 0(non unit in Z/3Z). Thus it isn't local homomorphisms.
I usually struggle to find examples on my own. Any suggestions, how to improve it?
How about $R=\{a/b:a,b\in\Bbb Z, b\text{ odd}\}$ where the maximal ideal is $2R$ and $S=\Bbb Q$. Then the inclusion $R\to S$ is not a local homomorphism.