Earlier I wrote this question about an algorithm computed on an integral domain.  However as commented I didn't suggest any particular ways of storing an integral domain in computer memory.  I set out to do this but found it a particularly difficult task.It is in fact impossible to store all the integral domains in memory because the fields are uncountable1 and are a subset of the integral domains.  However I am not concerned with fields because they don't have any primes anyway.How might one represent non-field integral domains in computer memory?  Is the set of all non-field integral domains countable?  If it is uncountable is there any useful subset of them that can be represented in computer memory?1: The rationals are a field.  Every polynomial extension of a field is also a field.  Thus the number of polynomial field extensions of $\mathbb{Q}$ is the powerset of the irreducible polynomials on $\mathbb{Q}$ which is uncountable.
Representing Integral domains in computer memory?
group theory