How to prove a set is an integral domain?

Question

Let's say I have the ring: $\mathbb{Z}[\sqrt{2}]=\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$.

Now the question asked is to prove whether or not this ring is an integral domain.

By definition: "An integral domain is a commutative ring with unity and no zero-divisor"

From this I would try to prove three things:

1. ring is commutative
2. ring has a unity
3. ring has no zero-divisors

Yet on multiple occasions and multiple examples only the last one (no zero-divisor is proven), why is this the case? Is there no need to prove the other two?

Answer 1

To prove that it is an integral domain, you would indeed have to prove all three. That it has a unity and is commutative is quite obvious, but that it has no zero-divisors may not be immediately obvious.

Fortunately, all three follow from the fact that it is a subring of the real numbers, which is an integral domain.

Answer 2

These "multiple occasions" may be implicitly applying the subring test, i.e. $\,\Bbb Z[\sqrt 2] \subseteq \Bbb R\,$ contains $\,1_{\Bbb R}$ and is easily verified to be closed under subtraction and multiplication hence it is a subring of $\,\Bbb R.\,$ Furthermore, it is clear that nontrivial subrings of domains remain domains (since the inference$\,a,b\neq0\,\Rightarrow\, ab\neq 0 $ necessarily remains true in every subring containing $\,a,b)$

Remark $ $ In fact, by the general definition of ring adjunction, $\,\Bbb Z[\sqrt 2]\subseteq\Bbb R$ is the intersection of all subrings of $\Bbb R$ containing $\,\Bbb Z\,$ and $\,\sqrt 2\,$ so it is a domain, being a nontrivial intersection of domains.

Such inferences are quite common in algebra, because we define (minimally) "generated" structures via such intersections, and the (universal) logical form of the axioms of many algebraic structures makes it immediately clear that they are closed under intersections and subalgebras (e.g. the $\rm\color{#c00}{universal}$ form ring axiom $\,\color{#c00}{\forall x,y:}\ x+y = y+x).\,$ This is made more precise when studies the relationship between syntax and semantics in universal algebra and model theory.