# Thread: 2 Ring Isomorphism Questions

1. ## 2 Ring Isomorphism Questions

1.) Determine all ring isomorphisms from Zn to itself.

2.) Determine all ring homomorphisms from Z+Z into Z+Z (where + is the Direct Product).

For #1, I know that Zn has unity 1, so phi(1) = 1. Does that mean Phi(x) = x is the only ring isomorphism from Zn to itself?

I'm really stuck on #2. Any help would be greatly appreciated.

Thanks in advance!

2. I may have gotten #2. Are there 9 different ring homomorphisms that work? Thanks.

3. Originally Posted by GB89
For #1, I know that Zn has unity 1, so phi(1) = 1. Does that mean Phi(x) = x is the only ring isomorphism from Zn to itself?
If $\displaystyle \phi(1) = 1$ then $\displaystyle \phi(n) = \phi(1+...+1) = \phi(1) + ... + \phi(1) = n\phi(1) = n$.

Hint: If you know what $\displaystyle \phi(1,0)$ and $\displaystyle \phi(0,1)$ is then you can determine everything.

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# all ring isomorphism from zn to itself

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