Group and ring isomorphism

Question

Why is the group $2\mathbb Z$ (group of even integers) under addition, group-isomorphic to the group $\mathbb Z$ under addition, whereas, the ring $2\mathbb Z$ is not ring-isomorphic to the ring $\mathbb Z$?

Answer 1

Well, multiplicatively the ring $2{\Bbb Z}$ has no unity, i.e., no element $e$ such that $e\cdot z = z = z\cdot e$ for each $z\in 2{\Bbb Z}$, whereas the ring of integers $\Bbb Z$ has.

Answer 2

More generally, consider a multiplicative map $f: \mathbb Z \to 2\mathbb Z$, in the sense that $f(xy)=f(x)f(y)$.

Let $f(1)=2a$. Then $2a=f(1)=f(1\cdot 1)=f(1)f(1)=4a^2$ implies $a=0$ or $2a=1$. Since $a\in\mathbb Z$, we have $a=0$. Then, $f(n)=f(1\cdot n)=f(1)(n)=0$.

Bottom line: the only multiplicative map $f: \mathbb Z \to 2\mathbb Z$ is the zero map.