# Thread: Homomorphism, kernel, monomorphism :)

1. ## Homomorphism, kernel, monomorphism :)

Given $f$, natural mapping from $\mathbb{Z} \text{ to }\mathbb{Z}_n\text{ with }f(m)=r$, where $r$ be the remainder if $m$ is divided by $n$.
Show that $f$ is homomorphism!
And determine $ker(f)$.
Is $f$ monomorphism?

2. Originally Posted by GTK X Hunter
Given $f$, natural mapping from $\mathbb{Z} \text{ to }\mathbb{Z}_n\text{ with }f(m)=r$, where $r$ be the remainder if $m$ is divided by $n$.
Show that $f$ is homomorphism!
And determine $ker(f)$.
Is $f$ monomorphism?
rtp $f(ab)=f(a)f(b)$. $f(a)f(b)=rs$, by definition.

Now, expand $ab=(xn+r)(yn+s)$ to get the other side of the equality.

What do you think the kernel of such a mapping should be? Remember, the kernel is everything that is mapped to zero.

What is $f(a)$ for $1 \leq a \leq n$? What does this tell you about surjectivity?