Proof check: find all Prime ideals of $R[T]/\langle x^n\rangle$,

Question

Given $R=\mathbb{C}[x]/\langle x^n\rangle$, each element in $R$ may be represented as $a_0+a_1x+\cdots+a_{n-1}x^{n-1}$. I'm guessing that $P=\langle x^i\rangle$ for $i=\{1,\ldots,(n-1)\}$ represents all possible primes in $R$. Am I correct or are there aditional prime ideals that I'm missing.

Answer 1

Those are not prime, unless $i=1$. In fact, $k[X]/(X^n)$ is a local ring with maximal nilpotent principal ideal $(\bar X)$, hence with finitely many ideals. Since the maximal ideal is nilpotent, it is the only prime ideal: $(\bar X)^n=0\subseteq \mathfrak p$ implies $(\bar X)\subseteq \mathfrak p$ implies $(\bar X)=\mathfrak p$ by maximality.