Showing that rings kZ and lZ, where l≠k are not isomorphic.

[jmjlt88]

I have shown that Z is not isomorphic to 2Z and that 2Z is not isomorphic to 3Z. I need now to generalize this. Thus, to prove that rings kZ and lZ, where l≠k are not isomorphic, I need to define an arbitrary isomorphism, and reach a contradiction. So here's what I am thinking. I let f: kZ->lZ be a isomorphism. Then, f(k)=ln , where n is some integer. This is true since k in kZ has to go to some multiple of l. From here, I have just been tinkering around trying to get k to map to 0 so that f is not injective. This worked for the first two. Is this the right approach? Not really looking for a hint, but just an "okay" to keep trying working with this line of reasoning.

 

[micromass]

Yeah, I think that's a valid approach. Go on, you'll get there.

Ask anytime if you want a hint

 

[jmjlt88]

Okay, I give! =) A little hint would be great. When working with specific numbers, it was so easy to show that an isomorphism cannot exist. Now, I cannot get anything to work (or not work, I guess). A nudge in the right direction would be nice. Or else, this problem will keep me up all night!

 

[micromass]

Okay, I give! =) A little hint would be great. When working with specific numbers, it was so easy to show that an isomorphism cannot exist. Now, I cannot get anything to work (or not work, I guess). A nudge in the right direction would be nice. Or else, this problem will keep me up all night!

Think about f(k^2).

 

[jmjlt88]

Hmmmm...

I worked with that for while. Perhaps, I couldn't see the forest through the trees (or maybe I made an error). Here's what I had.

f(k²)=f(k)f(k)=l²n²

f(k k)=f(k)+f(k)+...+f(k) [k-times]=k(ln)

Now, I know that k≠l. But perhaps there would be a problem if k=ln?

 

[micromass]

OK, so we can deduce that k=ln.

Now think about the inverse map f^{-1} and do something similar.