There's a Lemma about monomial ideals that says:
- "Let $I=\left<x^α \mid \ α ∈ A\right>$ be a monomial ideal. Then a monomial $x^β$ lies in $I$ if and only if $x^β$ is divisible by $x^α$ for some $α ∈ A$."
My question here is: take for example a monomial ideal $I=\left<f_1,f_2,f_3\right>=\left<x^5y,y^5z,z^5x\right>$. If I want to know if a polynomial/monomial $g \in I$, it's enough to show that $g$ is divisible by ONE of the $f_s$ ($s=\{1,2,3\}$)?
Or does $g$ has to be divisible for ALL of the $f_s$?
A polynomial $g$ is in the ideal $I=\langle f_1, f_2, f_3\rangle$, if you can write $g = a_1 f_1 + a_2 f_2 + a_3 f_3$ with $a_i$ taken from the ring of polynomials.
If $g$ is divisible by one of the polynomials (say $f_1$) with quoutient $c$, it can be written as $g = cf_1$ rioght? Since $0$ is always a part of the ring, you can say $a_2 = a_3 =0$ and have a presentation, that gives you $g \in I$.
Keep in mind, that the conversion is not always true, Even, if $g$ is not divisible by $f_1$ to $f_3$ it can still be part of $I$. For that, read on Groebner bases.