Is $(x,y) \mapsto 0$ on $\mathbb{Q}\backslash\{0\}$ associative and commutative? I have the following definition of operations on the following sets:


*

*$(x,y) \mapsto 9xy$ on $\mathbb{Z}$

*$(x,y) \mapsto 0$ on $\mathbb{Q}\backslash\{0\}$


I have to determine whether the operations on the given sets are associative, commutative, have a neutral element, and have inverse elements.
For $(x,y) \mapsto 9xy$ I have that it is associative, commutative, and has the neutral element $1 \in \mathbb{Z}$, but does not have inverse elements as $(9xy)^{-1} \notin \mathbb{Z}$.
Could you please help me with $(x,y) \mapsto 0$? I don't understand the operation. It always maps $(x,y) \mapsto 0$, so how do I prove if this is associative, commutative etc.?

The first one is associative and commutative as these two identities are homogeneous and then 
$$
(x,y)\rightarrow qxy
$$
is such on $\mathbb{Z}\times \mathbb{Z}$ for all $q\in \mathbb{Z}$. Only $q\in \{-1,1\}$ provides neutral.
The second is not even internal.   
