Examples of $\mathcal{D}$-chains?

Question

I was reading through this paper and came across an interesting definition:

Let $\mathcal{D}$ denote a set of prime ideals of $S$. A chain $\mathcal{C}$ of prime ideals of $R$ is a $\mathcal{D}$-chain if $f^{-1}(Q) \in \mathcal{D}$ for every $Q \in \mathcal{C}$.

In the paper, $f:S \rightarrow R$ is a ring homomorphism with $f(S)$ contained in the center of $R$, where $f$ is not necessarily unitary and $R$ and $S$ are not necessarily commutative. Both $R$ and $S$ are nontrivial rings with identity.

The paper doesn't provide specific examples of $\mathcal{D}$-chains, but I'm having trouble manufacturing any. Are there any relatively easy-to-see examples of $\mathcal{D}$-chains?

I'd be happy with just one example, even if it assumes $S$ and $R$ are commutative or $f$ is the inclusion map, for example.

Thanks!

Answer 1

How about we say $R$ and $S$ are commutative, $f$ is onto, and $D$ is the set of all prime ideals of $S$. Then every chain of prime ideals in $R$ is a $D$ chain.

Presumably the interesting stuff happens when $D$ is a particular subset of primes.