How Do You Find All Ring Homomorphisms for Specific Mappings?

[phyguy321]

Homework Statement

Find all ring homomorphisms \phi: Z \rightarrow Z
\phi: Z₂ \rightarrow Z₆
\phi: Z₆ \rightarrow Z₂

Homework Equations

A function \phi: R \rightarrow S is called a ring homomorphism if for all a,b\inR,
\phi(a+b) = \phi(a) + \phi(b)
\phi(ab) = \phi(a)\phi(b)
\phi(1_R) = 1_S

The Attempt at a Solution

 

[Dick]

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

 

[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?

 

[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?

 

[Hurkyl]

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

 

[phyguy321]

Z₆ \rightarrow Z₂ \phi(a mod 6) = a mod 2. since if a \equivb mod 6 then a\equivbmod 2 since 2|6

 

[Dick]

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

 

[HallsofIvy]

Z_2 only has two members. Z_{6[/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!}

 

[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.

 

[phyguy321]

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

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 can't 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