Why is the affine $\Bbbk$-algebra, $ \Bbbk[x]/\langle x^3 \rangle $ zero-dimensional?

Question

Consider the ideal $\mathfrak{a} = \langle x^3 \rangle \subseteq \Bbbk[x]$. The ideal $\langle x + \mathfrak{a} \rangle$ is a prime ideal in $ \Bbbk[x]/\mathfrak{a}$. Then why is the affine algebra, $ \Bbbk[x]/\mathfrak{a}$ zero-dimensional?

Answer 1

The prime ideals of $k[x]/(x^3)$ are of the form $\mathfrak p/(x^3)$ where $\mathfrak p$ is a prime ideal in $k[x]$ containing $(x^3)$. But $k[x]$ is a principal ideal domain, so $\mathfrak p=(f)$ with $f\in k[x]$ irreducible. From $(x^3)\subseteq\mathfrak p$ we deduce $x\in\mathfrak p$, so $f\mid x$. It follows that $f=ax$ for some $a\in k$, $a\ne0$. Then $(f)=(x)$, and thus $(x)/(x^3)$ is the only prime ideal of $k[x]/(x^3)$.