Decide what we mean by "irreducible ring element" and define it in the manual (possible in the `is_irreducible` docstring)

[fingolfin]

Notice while working on PR #1952: we don't actually define what we mean by "irreducible element" in a ring. Now for a integral domain, there is a no debate, the well-known definition works:

$a\in R$ is irreducible iff $\forall b,c\in R: a=bc$ implies $b$ is a unit or $c$ is a unit.

But what about commutative rings with zero divisors? Wikipedia discussed several incompatible definitions for this case. The "usual" above is apparently also called "very strongly irreducible" in this context, while they quote a source that suggests irreducible should mean "whenever $a=bc$ then $(a)=(b)$ or $(a)=(c)$".

One perfectly reasonable answer to this of course would be to just say we only define and support is_irreducible, factor, etc. over domains "in general" (specific implementations of non-domains then are of course still free to support these functions, they then just need to specify which notion of "irreducible" they have in mind).

If were to pick a definition, we should be careful to pick one that we can actually compute, and that actually is of interest to anyone. In other words: If we don't have a concrete application, it is probably best to just restrict this to domains, so that we can revisit this in the future if we have an application.

[thofma]

My opinion is that we should not define it for non-domains. We can revisit if someone has a concrete application, where one can actually compute something.