Show that two rings are not isomorphic

[Mr Davis 97]

Homework Statement

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

Homework Equations

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

 

[phinds]

Homework Statement

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

Homework Equations

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"

 

[fresh_42]

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.

 

[nuuskur]

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?