My question is simple to ask: given some prime ideal $P$ in a ring $R$, we can talk about $P^2, P^3$ etc. but can we discuss $P^0$? Is there a convention that says $P^0 = R$, or is there something more concrete and computable than that?
If it makes any difference, I am specifically working with some prime ideal $\mathfrak{p}$ in the ring of integers $\mathcal{O}_K$ of some field $K$.
Thank you!
EDIT: (more information) It's in the context of discrete valuation, specifically the $\mathfrak{p}$-adic valuation $v_{\mathfrak{p}}$ defined to be the map which sends elements $x \in \mathcal{O}_K$ to the largest integer $i$ such that $x \in \mathfrak{p}^i$. I want to show that $\mathcal{O}_K$ is the valuation ring of $K$ w.r.t $v_{\mathfrak{p}}$, and all I require for this to be true is that $\mathfrak{p}^0 = \mathcal{O}_K$.
Here is a link to a textbook that adopts the convention $\mathfrak{p}^0 = R$ explicitly (see the bottom line).