Which elements of $\mathbb Z \times \mathbb Z$ are zero divisors?

Question

Which elements of $\mathbb Z \times \mathbb Z$ are zero divisors?

I am asked to look at the following example:
Let $R$ and $S$ be rings, and $R \times S$ the Cartesian product of $R$ and $S$, (r,s) with $r \in R$ and $s \in S$. then $R \times S$ become the ring with the following operations:
$(r_1, s_1)+(r_2,s_2)=(r_1+r_2, s_1+s_2)$
and $(r_1,s_1)(r_2,s_2)=(r_1r_2,s_1s_2)$ for all $r_1,r_2 \in R, s_1, s_2 \in S$

The only way to get zero for addition is by adding the elements inverse to itself, so $r_1+-r_1$ and the only way to get 0 in multiplication is by multiplying by 0. An element is called a zero divisor in r if there an element b, where b is not 0, such that $ab=0$ So would this be an integral domain then?

Answer 1

Hint: try to compute $(1,0)(0,1)$...

Answer 2

Simply take $(a,0)$ ($(0,a)$ respectively) with $a\neq 0$

One has $(a,0).(0,1)=(0,0)$ (respectively $(1,0).(0,a)=(0,0)$).