Homomorphism, kernel, monomorphism :)

**GTK X Hunter** · November 15th 2009, 09:00 PM

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?

**Swlabr** · November 16th 2009, 01:20 AM

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?

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