# Show that two rings are not isomorphic

• Mr Davis 97
In summary, to show that the rings ##2 \mathbb{Z}## and ##3 \mathbb{Z}## are not isomorphic, we can assume they are isomorphic and then find a contradiction by considering the consequences of this assumption. One way to do this is by trying to find a ring homomorphism between the two rings and see if we run into contradictions.

## Homework Statement

Show that the rings ##2 \mathbb{Z}## and ##3 \mathbb{Z}## are not isomorphic.

## The Attempt at a Solution

I know how to show that two structures are isomorphic: find an isomorphism. However, I am not quite sure how to show that there exists no isomorphism at all

Mr Davis 97 said:

## Homework Statement

Show that the rings ##2 \mathbb{Z}## and ##3 \mathbb{Z}## are not isomorphic.

## The Attempt at a Solution

I know how to show that two structures are isomorphic: find an isomorphism. However, I am not quite sure how to show that there exists no isomorphism at all
I don't have any help for you but I have to comment that your post gave me a chuckle because my son and I were just recently discussing a linguistic phenomenon that I noticed some time ago. I make no representation that I know what YOU mean, but I know what I mean, and what most people mean with the following construct:

"I don't exactly know how to ... " or "I don't quite know how to ... " generally means "I don't have even the tiniest clue how to and in fact I'm not even sure how to spell it"

nuuskur and Mr Davis 97
Assume they were isomorphic and consider the consequences, or try to find a ring homomorphism by defining ##\varphi (2) = 3x## for some integer ##x## and see if you run into contradictions.

Suppose, for a contradiction, a ring isomorphism $f :2\mathbb{Z}\to3\mathbb{Z}$ existed. Then $f(2) = 3m$ for some $m\in\mathbb{Z}$.
Since $f$ respects addition and multiplication, then $f(2)+f(2) =f(2+2) =f(4)= f(2\cdot 2)=f(2)\cdot f(2)$. But this is a problem. Can you explain, why?

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Mr Davis 97