# Thread: Homomorphism, kernel, monomorphism :)

1. ## Homomorphism, kernel, monomorphism :)

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

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

Now, expand $\displaystyle ab=(xn+r)(yn+s)$ 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 $\displaystyle f(a)$ for $\displaystyle 1 \leq a \leq n$? What does this tell you about surjectivity?