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.

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*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.)
