Define $ker(f)$ for $f:(\mathbb{Z},+)\to (\mathbb{Z}/_{12},+)$

Question

I need to define the $ker(f)$ for the following homomorphism

$f:(\mathbb{Z},+)\to (\mathbb{Z}/_{12},+)$

Where $f(x) = [x]_{12}$

If I understood that correctly then the $ker(f) = \{a \in \mathbb{Z} \mid f(a) = e\}$ And therefore the solution is

$$ker(f) = \{...,-12,0,12,...\}$$

As the identity element for a coset with addition should be $[0]_{12}$

Is that correct?

Answer 1

Is that correct?

Yes. $\mathrm{ker}(f) = \{12k : k \in \mathbb{Z}\}$.