Is there a finite dimensional local ring with infinitely many minimal prime ideals?
Equivalent formulation:
Is there a ring with a prime ideal $\mathfrak p$ of finite height such that the set of minimal prime sub-ideals of $\mathfrak p$ is infinite?
Here "ring" means "commutative ring with one", "dimension" means "Krull dimension", and "local ring" means "ring with exactly one maximal ideal" (warning: some authors call "quasi-local ring" a ring with exactly one maximal ideal, and "local ring" a noetherian ring with exactly one maximal ideal; it is well known that a noetherian ring has only finitely many minimal prime ideals).
Best Answer
Let $R$ be the quotient of a polynomial ring $k[x_1,x_2,\dots]$ in infinitely many variables over a field by the ideal generated by all products $x_ix_j$ for $i\neq j$. Geometrically, $\operatorname{Spec} R$ looks like infinitely many copies of $\mathbb{A}^1_k$ with their origins identified. So, $R$ is a $1$-dimensional ring with infinitely many minimal prime ideals, each of which are contained in the maximal ideal generated by all the $x_i$. See this answer for more details about this ring.