# Homework Help: 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