Homomorphism, kernel, monomorphism :)

• November 15th 2009, 09:00 PM
GTK X Hunter
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!(Wondering)
And determine $ker(f)$.(Itwasntme)
Is $f$ monomorphism?(Giggle)
• November 16th 2009, 01:20 AM
Swlabr
Quote:

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!(Wondering)
And determine $ker(f)$.(Itwasntme)
Is $f$ monomorphism?(Giggle)

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?