Finding Ring Homomorphisms

1. Oct 22, 2008

phyguy321

1. The problem statement, all variables and given/known data
Find all ring homomorphisms $$\phi$$: Z $$\rightarrow$$ Z
$$\phi$$: Z2 $$\rightarrow$$ Z6
$$\phi$$: Z6 $$\rightarrow$$ Z2

2. Relevant equations
A function $$\phi$$: R $$\rightarrow$$ S is called a ring homomorphism if for all a,b$$\in$$R,
$$\phi$$(a+b) = $$\phi$$(a) + $$\phi$$(b)
$$\phi$$(ab) = $$\phi$$(a)$$\phi$$(b)
$$\phi$$(1R) = 1S

3. The attempt at a solution

2. Oct 22, 2008

Dick

So why is that difficult for you? You have to show an attempt or state what is confusing you before anyone can help.

3. Oct 23, 2008

phyguy321

so i have to find every set in Z that satisfies those equations by ending in Z?
same goes for Z_2 to Z_6 find every set that will add together in the homomorphism in Z_2 and will separately add together in Z_6? is this what its asking?
if so how do i show that?

4. Oct 23, 2008

Dick

Your definition says phi(1)=1. Can you use that with the other homomorphism properties to figure out what phi(k) must be for the other k's in the domain ring?

5. Oct 23, 2008

Hurkyl

Staff Emeritus
A journey of infinite length starts with a single step....

6. Oct 23, 2008

phyguy321

Z6 $$\rightarrow$$ Z2 $$\phi$$(a mod 6) = a mod 2. since if a $$\equiv$$b mod 6 then a$$\equiv$$bmod 2 since 2|6

7. Oct 23, 2008

Dick

The answer is correct. But I can't say the reason really captures the what the problem is about.

8. Oct 24, 2008

HallsofIvy

Z_2 only has two members. Z6[/sup] only has 6 members. It shouldn't be all that hard to write down all functions from Z2 to Z6 much less just all homomorphims were you know 0Z2---> 0Z6!

9. Oct 24, 2008

enigmahunter

Z is the initial object of category of rings with morphism f:Z->S satisfying f(1z) = 1s (1z is the mulitplicative identity of Z and 1s is the multiplicative identity of a ring S.

That means, a ring homomorphism f from Z to any ring is unique as long as f:Z->S satisfying f(1z) = 1s.

Last edited: Oct 24, 2008
10. Oct 27, 2008

phyguy321

frankly I don't really care about capturing the reason of the problem. I just need to get through this class and not have a W on my transcript. Abstract math and modern algebra are terrible awful aspects of math that i just cant grasp.
so as long as that is something i can put down and get credit for I don't care, I'll never have to do it again