In these references, Coquand, Lombardi, and Roy give an elementary characterization of Krull dimension, which inductively makes use of one of two notions of the "boundary" of a subvariety, given as follows:

Let $R$ be a commutative ring, and $x\in R$.

$$\text{upper boundary} R^{\{x\}} := R/I^{\{x\}},\,\,\, I^{\{x\}} := xR + (\sqrt{0}:x)$$ $$\text{lower boundary} R_{\{x\}} := S_{\{x\}}^{-1}R,\,\,\, S_{\{x\}} := x^{\mathbb{N}}(1+xR)$$

where $(\sqrt{0}:x)$ is the ideal quotient of the nilradical, and $x^{\mathbb{N}}(1+xR) = \{x^n(1+rx) | n\in\mathbb{N}, r\in R\}$.

Clearly, $\mathrm{Spec}(R^{\{x\}})$ is closed containing the locus $V(x)$, and $\mathrm{Spec}(R_{\{x\}})$ is a localization (not quite open) that is disjoint from the locus $V(x)$. Also, both are trivial exactly when $x\in R^\times \cup \sqrt{0}$.

However, I do not have good intuition for these subschemes.

1. How to think about these boundary schemes? Do they represent anything in particular?

2. Do these constructions appear anywhere else in the literature? I have not been able to find anything.

3. Are they commutative, in that $R^{\{x\}\{y\}} = R^{\{y\}\{x\}}$ and $R_{\{x\}\{y\}} = R_{\{y\}\{x\}}$?

4. Are these very natural constructions? I.e. would it be worth studying them in more detail, in specific cases, or are they primarily instrumental in the characterization of Krull dimension?