You don't really have to do all of that. To show that a ring is an integral domain, you just need to show that it's commutative, have unity, and have no zero divisors.

1) Let and . Then for some , which means that or . Therefore, in or .

2) is a commutative ring with unity; therefore, we need to show that there are no zero divisors.

Let , where

,

and and . Then has leading coefficient , but since is an integral domain, .