If there are any missing hypotheses without which the things discussed in this question make no sense, please let me know.
Let $R$ be a ring with norm $\mathfrak{N}$ defined by $\mathfrak{N}\left(\mathfrak{a}\right)=|R/\mathfrak{a}|$ and define $$\zeta_{R}\left(s\right)=\sum_{\mathfrak{a}}\mathfrak{N}\left(\mathfrak{a}\right)^{-s},$$
where the sum is over all non-zero ideals of $R$.
Letting $k,r\geq1$, not both 1, we say that the ideals $\mathfrak{a}_{1},\ldots,\mathfrak{a}_{k}\subseteq R$ are relatively $r$-prime if there is no nonzero ideal $\mathfrak{b}\subseteq R$ with $\mathfrak{a}_{i}\subseteq \mathfrak{b}^{r}$ for all $i=1,\ldots,k$. Setting $k=1$ gives $r$-free and, on the other hand, setting $r=1$ gives relatively prime.
It is known that the probability/natural density of such ideals is $\displaystyle{\frac{1}{\zeta_{R}\left(rk\right)}}$ for $R=\mathcal{O}_{K}$ [1] and $\mathbb{F}_{q}\left[x\right]$ [2], where $\mathcal{O}_{K}$ is the ring of integers of an algebraic number field $K$ (with $K=\mathbb{Q}$ giving the classical setting), and $q$ is any prime power.
My question is: Are there other rings for which this (or similar) holds? If so, what are the relevant papers?
[1] Sittinger, Brian, The probability that random algebraic integers are relatively $r$-prime. J. Number Theory 130 (2010) 164-171.
[2] K. Morrison, Z. Dong, The probability that random polynomials are relatively $r$-prime, 2004. Link: http://www.calpoly.edu/~kmorriso/Research/RPFF04-2.pdf