2

User fbakhshi deleted the following question:

Let $G$ be a simple graph with finite vertex set $X = \{x_1,\dots, x_n\}$. The edge ideal of $G$, denoted by $I = I(G)$, is the ideal of $R=K[x_1,\dots,x_n]$ generated by all square-free monomials $x_ix_j$ such that $\{x_i, x_j\} \in E(G)$.

Prove that $\operatorname{Min}(I)=\operatorname{Ass}(I)$, where $\operatorname{Min}(I)$ and $\operatorname{Ass}(I)$ are the minimal prime ideals and associated prime ideals of $I$ respectively.

Community
  • 1

1 Answers1

0

The property holds more generally for square-free monomial ideals, not only for edge ideals.

It's easy to see that the associated ideals of $I$ are generated by subsets of the variables set $\{x_1,\dots,x_n\}$ and this is enough.