It is clear that for any $n$ there is a ring $R$ with exactly $n$ prime ideals (e.g., consider a product of $n$ fields). Can one construct such a ring whose prime ideals form a chain?

I can show that such rings do exist: Let $X$ be the set with $n$ points $x_1$, ..., $x_n$ and define a topology on $X$ with open sets $\emptyset$, $X$, $\{x_1\}$, $\{x_1,x_2\}$, ..., $\{x_1,...,x_n\}$. Then $X$ with this topology is a spectral space, which means that there exists a ring $R$ such that $Spec(R)$ is homeomorphic to $X$. Then clearly this $R$ has exactly $n$ prime ideals and they form a chain.