# Homomorphism, kernel, monomorphism :)

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

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

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?