How can I find the neutral element of this algebraic structure

Question

Given the algebraic structure $(Z×Z,⊥)$, with

$(a,b)⊥(c,d)=(a+c,bd)$

for each $a,b,c,d∈Z$.

How can I find the neutral element of $(Z×Z,⊥)$? Please help me.

Answer 1

Let $e=(c,d).$

Thus, $a+c=a$ and $bd=b$ for all $b\in\mathbb Z$, which gives $c=0$ and $d=1$.

Easy to see that for all $x$ from our structure $$x\perp e=e\perp x=x.$$