Let $k$ be a finite field of order $q$. Let $x$ be a closed point in $\mathbb{P}^1_k$ and let $D$ be a divisor $D = (e+1) x$, where $e \in \mathbb{N}$.
We define few things, $$ P_N = \{ x \in k[u,v]: \text{homogeneous and of degree } N \}, $$ $$ O_D = P_N / \Gamma(\mathbb{P}^1, O(N)(-D)), $$ where I understand $\Gamma(\mathbb{P}^1, O(N)(-D))$ to be the forms in $P_N$ that vanish on $D$, and $$ R_e = k[t]/(t^{e+1}). $$
I think $O_D$ is isomorphic to $R_e$ as a $k$-algebra...
Could I possibly verify with someone if this is the case or not? If so, I am interested in finding out how to show it. (If not is it true for $R_a$ for some $a \in \mathbb{N}$?) I would appreciate any reference or hint or explanation!
Suppose $x$ is a $k$-rational point of $\mathbb P^1_k$ (and not only a closed point!)
The effective divisor $D=(e+1)x$ corresponds to a non-reduced closed subscheme $X_D\subset \mathbb P^1_k$ with support $x$.
The $k$-algebra of global functions of that scheme $X_D$ is indeed given for all $e\geq 0$ by $\Gamma(X_D,\mathcal O_{X_D})=k[t]/(t^{e+1})$ .
The introduction of $N$ is an irrelevant distraction: if you are interested in another question, I would suggest you ask it in a new post.