Dirichlet rings Dirichlet's theorem on arithmetic progressions says that given coprime $a,b\in \mathbf Z$, there are infinitely many positive integers $n$ such that $a+bn$ is prime.
Let us call a ring $R$ a Dirichlet ring if it is a domain of characteristic $0$ and we have the following:

For any $a,b\in R$, if $R=aR+bR$, then there are infinitely many $n\in \mathbf N$ such that $a+nb$ is prime in $R$.

Let us also call $R$ weakly Dirichlet if the above holds for $a,b\in \mathbf Z$.
How big is the class of (weakly) Dirichlet rings (e.g. does it include all Dedekind domains)? Is there any other useful characterisation of either class?
My guess is that a ring of integers in a number field is Dirichlet, but my knowledge of number theory is rather superficial, so I'm not very confident even about that. It is easy to see that being weakly Dirichlet implies that infinitely many prime integers remain prime in $R$, which should restrict the range of suspects somewhat.

Addendum: The variant where there are infinitely many $r\in R$ with $a+rb$ prime is also interesting.
 A: An example of a Dedekind domain that is not weakly Dirichlet in your sense would be $R=\mathbb{Q}[X]$, as you can see by taking $a=b=1$. The picture would look different if you allow $n\in R$ instead of $n\in\mathbb{N}$, and there are some nontrivial in the literature in that direction.
EDIT: If we allow $n\in R$, here are a few results regarding polynomial rings $R=K[X]$ over a field $K$:
If $K$ is finite, $R=K[X]$ is a Dirichlet ring in that sense by a classical result of Kornblum, see e.g. Rosen, Number Theory in Function Fields, Section 4.
If $K$ is pseudo-algebraically closed but not separably closed, $R=K[X]$ is a Dirichlet ring by Theorem A of Bary-Soroker, Dirichlet's theorem for polynomial rings.
If $K$ is Hilbertian, e.g. a number field, $R=K[X]$ is easily seen to be a Dirichlet ring in that sense (see e.g. the introduction of the paper by Bary-Soroker).
If $K=\mathbb{C}((t))$, $R=K[X]$ is not a Dirichlet ring in that sense. I'm not sure if this is written anywhere, but I could look it up or sketch a proof.
