# Cellular and primary binomial ideals

Let $$I \subseteq \mathbb{K}[x_1, \dots, x_n]$$ be an ideal of a polynomial ring over a field $$\mathbb{K}$$.

$$I$$ is called cellular if every variable $$x_i$$, with $$i=1, \dots, n$$, is either a nonzerodivisor modulo $$I$$ or is nilpotent modulo $$I$$.

$$I$$ is called primary if whenever $$fg \in I$$ then or $$f \in I$$ or $$g$$ is nilpotent modulo $$I$$.

Of course, if $$I$$ is primary $$\Rightarrow$$ $$I$$ is cellular.

My question is: consider a binomial ideal $$I \subset \mathbb{K}[x_1, \dots, x_n]$$, generated by binomials of the form $$f_k=\prod_{i \in I_k} x_i - \prod_{j \in J_k} x_j$$, for some $$I_k,J_k \subset \{1, \dots, n\}$$ with $$I_k \cap J_k = \emptyset$$ and both no empty.

What can I say about $$I$$ cellular $$\Rightarrow$$ $$I$$ primary? Under which hypothesis a cellular binomial ideal is primary?

Thanks

• The following two papers might be relevant: arxiv.org/abs/alg-geom/9401001 and arxiv.org/abs/0803.3846 Oct 27, 2018 at 17:12
• Thanks @AviSteiner. I knew them, but I don't find any help to solve my question. Oct 29, 2018 at 10:03
• @EllaSmith - Is the assumption that the ambient ring of $I$ is a polynomial ring over a field? (I am having trouble making sense of the phrase "every variable" in the definition of a cellular ideal.) Nov 3, 2018 at 1:19
• @benblumsmith I edited my question in order to answer to your comment. Nov 5, 2018 at 10:56
• Thank you @EllaSmith, this is what I needed to know. Nov 5, 2018 at 16:00

Consider the ideal $$I=\left\langle x_{1}^{3} x_{3}-x_{1}^{3}, x_{1}^{4}, x_{1}^{2} x_{2} x_{4}-x_{1}^{2} x_{2}, x_{2}^{2}, x_{4}^{3}-1\right\rangle \subseteq \mathbb{k}\left[x_{1}, x_{2}, x_{3}, x_{4}\right]$$ in Example 2.9 of this link. This is cellular but Macaulay2 says that it is not primary.