What are all the integral domains that are not division rings? A commutative division ring is an integral domain. But what are all the integral domains that are not division rings?
The examples I currently know are the following: $\mathbb{Z}$, $\mathbb{Z}[i]$, $\mathbb{Z}[\sqrt 2]$, $\mathbb{Z}[\sqrt k]$ where $k$ is not a perfect square, and the ring of polynomials $R[x]$ over any integral domain $R$. But what are all other examples?
Edit: I am a self-learner, so excuse me if the question is trival or stupid.
 A: As alluded to in the comments, since integral domains can be realized as subrings of fields, one could characterize integral domains as "the class of subrings of fields."
The integral domains that aren't fields are just such subrings that don't happen to be fields. This is the most straightforward classification you can hope for.
If you are still looking for more ways to produce integral domains, you should be aware of this one: $R/P$ is an integral domain for any commutative ring $R$ and prime ideal $P$ of $R$. This actually covers the first four examples you gave using the integers, since they are various quotients of $\Bbb Z[x]$ by prime ideals.
Since you mentioned division rings, I'll say something about noncommutative domains as well. The question is significantly harder here, since there exist domains that can't be embedded in division rings. For a completely prime ideal $P$ of $R$, meaning that $P$ satisfies the commutative definition of "prime ideal," $R/P$ is still a domain. However, completely prime ideals are harder to come by in noncommutative algebra than commutative algebra. It is also still true that polynomial rings over noncommutative rings are still domains.
A: *

*For any integral domain $D$, the polynomial ring $D[x_1, \ldots, x_k]$, $k > 0$.

*For any connected, open subset $U \subseteq \mathbb{C}$, the ring of holomorphic functions on $U$. (Note that the space of merely smooth functions on $U$ is not an integral domain.)


(Since all finite domains are fields, all examples of such rings are infinite.)
