Structural differences between $2 \mathbb{Z}$, $3 \mathbb{Z}$ as rings.

Question

Is usual to find in abstract algebra books this exercise:

Show that $2 \mathbb{Z}$ and $3 \mathbb{Z}$ are isomorphic as groups (with usual sum) but they aren't as rings (with usual sum and addition).

Its solution is based in consider, for example, the images f(4)=f(2+2) and f(4)=f(2*2) and conclude the "non existence" of such an isomorphism.

But, my question is, what are the structural differences (ring properties) that avoid this equivalence as rings?

Thanks in advance.

Answer 1

For such a homomorphism: $f(2)+f(2) = f(2+2) = f(4) = f(2*2) = f(2)*f(2)$.

Here $f(2) = 3*m$ for some nonzero integer $m$.

Thus $3*m + 3*m = (3*m)*(3*m)$, i.e., $2*3*m = 3^2*m^2$. Shortening gives $2 = 3*m$ which is a contradiction.