Associated Primes of Square free monomial ideals

Question

Let $I$ be a square free monomial ideal in $R=k[x_1,\dots,x_d]$. If $I$ is the edge ideal of a graph $G$, then the associated primes of $I$ are known.

My question is:

If $I$ is generated by square free monomials, are the associated prime ideals always of the form $(I:a)$ for some monomial $a$? Or if $P\in \mathrm{Ass}(R/I)$ then is it true that $P=(0:a)$ for some monomial $a$?

Answer 1

Yes, they are. The result holds for associated primes of graded modules (see Bruns and Herzog, Lemma 1.5.6(b)(ii)), and monomials are the homogeneous elements in some grading on $k[x_1,\dots,x_n]$.