Let i be the imaginary unit of $\Bbb{C}$ and $\mathbb{Z}[i]:= \{a + ib, ab \in\mathbb{Z} \} $
So the task is to prove that this a ring regarding multiplication and addition in $\mathbb{C}$ which is commutative, free of zero divisors and posseses a 1 element.
I already found out that it is a ring and that the 1 Element is (1,0) and 0 Element is (0,0), so in both cases $i= 0$. So I guess this is like proving a trivial Ring.
I am just confused because I am not sure about the $i$. How would I complete the proof? I am stuck because I don't know how to prove that it is free of zero divisors and not a field.
Thank you for the help.
Best Answer
The map $\Bbb Z[i]\to \Bbb Z/2\Bbb Z$, $a+bi\mapsto a+b+2\Bbb Z$ is a non-trivial ring homomorphism with non-trivial kernel. Such a thing does not exist for fields.