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:
- ring is commutative
- ring has a unity
- 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?
Best Answer
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.