Integral domain $0$ element

Question

An integral domain is defined as a commutative ring with 1 that it's elements comply $x*y=0 \Rightarrow x=0 $ or $y=0$

Is this element $0$ the one of $\Bbb Z$ or is the neutral element of the set we are working in such ring.

Answer 1

The latter: unless the ring is itself the integers, 0 will mean that ring's additive identity rather than the natural number 0.

And anyways, multiplication is a binary operation $R \times R \to R$, so unless 0 is an element of the ring, it won't be the product of two elements.