Representing Integral domains in computer memory?
In general this question has no precise answer: the class of integral domains is not even a set, much less a countable set! (For example, for each set $A$, we can find an integral domain $D_A$ larger than $A$, e.g. $\mathbb{Z}[A]$, which is sufficient to show that integral domains can't be all gathered into a set).

In general you need to specify something to describe the elements of the set, and how to compute multiplication and addition.

A popular choice, which covers a large portion of the interesting examples, is to take a known ring or field $R$, typically $\mathbb{Z}$ or $\mathbb{Q}$, or some finite field $\mathbb{F}_q$, and look at some quotient ring of the form $D = R[X_1,\ldots,X_n]/(P_1,\ldots,P_k)$, where the $P_i$ are polynomials in the $X_j$.

Now it's not even clear that $D$ is an integral domain, which is equivalent here to the fact that $I=(P_1,\ldots, P_k)$ is irreducible. This can be determined using the Gröbner basis method as explained here.

Now there is an obvious representation for the elements of $D$: every element is represented by a polynomial, which we know how to represent, and do addition and multiplication on. The only issue is equality, since it needs to be modulo $I$. But this also is possible using known techniques, similar to the Gröbner basis ones.

It would be great if someone had implemented all of this up somewhere, though. Oh but they have! The Sage mathematical software toolchain builds on all of these observations to offer a ton of ways to concretely work with these kinds of rings. They even have an is_integral method for quotient rings!

Sage is a fantastic piece of software that I highly recommend, which probably does a lot of what you want.