Stanley–Reisner ring

Remarks

The ideal $I_\Delta$ is already generated by the $x_F$ for the minimal non-faces $F$. It is a squarefree monomial ideal, and $\Delta \mapsto I_\Delta$ is a bijection from simplicial complexes on $[n]$ to squarefree monomial ideals of $S$. The Krull dimension of $k[\Delta]$ is $\dim \Delta + 1$, the largest size of a face.

Example: let $\Delta$ be the boundary of a triangle, with faces all subsets of $\set{1, 2, 3}$ except $\set{1, 2, 3}$. The only non-face is $\set{1, 2, 3}$, so $I_\Delta = (x_1 x_2 x_3)$ and $k[\Delta] = k[x_1, x_2, x_3]/(x_1 x_2 x_3)$, of Krull dimension $2$.

Non-example: the ideal $(x_1^2)$ of $k[x_1, x_2]$ is a monomial ideal, but it is not of the form $I_\Delta$, because it is not generated by squarefree monomials. Its quotient ring has the nonzero nilpotent element $x_1$, while every $k[\Delta]$ is reduced.