Finding the kernel of a homomorphism I have the groups of nonzero complex numbers and the positive real numbers and the homomorphism $f: \Bbb{C}^{*} \to \Bbb{R}_+$ such that $f(z)= \lvert z \rvert$. I need to find the kernel of f.
Now... $\ker(f)=\{f(z)=0\}$. I know that if $z=a+bi$, then $|z|= \sqrt{a^2+b^2}$ which is $0$ only if $a=b=0$, but then $z$ will also be $0$. What am I missing?
[EDIT]
Yes, silly me... my kernel was wrong, it should be $Ker(f)=${${\forall z \in C^*| f(z)=e=1}$}. Thank you all. Delete the question, maybe?

The kernel of $f$ is the set $\{z \mid f(z) = e\}$, where $e$ is the identity of $\Bbb R_{>0}$.
Now note that $\Bbb R_{>0}$ is a multiplicative group.

Further hint: So $e$ is a number satisfying $e \cdot x = x = x \cdot e$ for all $x \in \Bbb R_{>0}$...
