homomorphism from ZxZ to Q I have this problem. I like to find ring homomorphism from $(\mathbb Q,+,-)$ to ($\mathbb Z\times \mathbb Z,+,-)$ and
homomorphism from $(\mathbb Z\times \mathbb Z,+,-)$ to $(\mathbb Q,+,-)$.
My solution:
Let f is homomorphism. Suppose that $f(1)=(a,b)$, with $a,b∈\mathbb Z$. Since $f$ is a ring homomorphism it follows that in particular, $f(q)=f(q⋅1)=q⋅f(1)=q(a,b)$, with $q∈\mathbb Q$. 
That is all I did, becauseI do not know the next step. 
Many thanks.
 A: Apart from $x \mapsto (0, 0)$ (if you allow that as a ring homomorphism), no such ring homomorphism can exist from $\Bbb{Q}$ to $\Bbb{Z}\times\Bbb{Z}$, because if $f$ were a ring homomorphism from $\Bbb{Q}$ to $\Bbb{Z}\times\Bbb{Z}$, then $f(1/2)$ would be an element, $x$ say, of $\Bbb{Z}\times\Bbb{Z}$ such that $2x = (1, 1)$, which is impossible.
As for the oppositve direction, $(x, y) \mapsto x$ and $(x, y) \mapsto y$ both define ring homomorphisms from $\Bbb{Z}\times\Bbb{Z}$ to $\Bbb{Z}\subseteq\Bbb{Q}$.
A: Not only is there no ring homomorphism, but there is no nontrivial group homomorphism $\Bbb Q\to\Bbb Z\times\Bbb Z$, and there are only two nontrivial ring homomorphisms from $\Bbb Z\times\Bbb Z$ to $\Bbb Q$.
The group $(\Bbb Q,+)$ is divisible, that is, for every $g\in\Bbb Q$ and every nonzero $n\in\Bbb Z$, there is $g'\in\Bbb Q$ such that $ng'=g$. However, you can easily see that the image (under a homomorphism) of a divisible group must be divisible. The only divisible subgroup of $\Bbb Z\times\Bbb Z$ is the trivial subgroup.
An idempotent $e$ in a ring $R$ is an element such that $e^2=e$; a homomorphism must take idempotents to idempotents. The ring $\Bbb Z\times\Bbb Z$ has four such elements, namely $(0,0)$, $(1,0)$, $0,1)$, and $(1,1)$. The answer of Rob Arthan correctly identifies the homomorphisms that take $(1,0)$ and $0,1)$ to $1\in\Bbb Q$; any homomorphism taking $(1,1)$ to $1\in\Bbb Q$ must be one of these, since if both go to $1$, then $(1,1)\mapsto2\in\Bbb Q$, not an idempotent. So there are only two nontrivial ring homomorphisms from $\Bbb Z\times\Bbb Z$ to $\Bbb Q$.
A: $Q(p/q) \to  Z(p,q)$
$p/q+r/s = (ps + qr)/qs$
The addition operation in $\mathbb Z\times \mathbb Z$
has to be defined as: $(p,q) + (r,s) = (ps+qr,qs)$
