Criterion for Irreducible Monomial Ideals I am working on the following problem:

Show that (a monomial ideal $I \subset K[x_1, ... , x_n]$ cannot be written as the intersection of two strictly larger monomial ideals) if only if ($I$ has a generating set consisting of powers of variables).

To be honest, I don't really know where to start!  I don't particularly have any immediate thoughts on showing either implication, and neither way seems particularly obvious to me.  
I'm looking for a nudge in the right direction really; a starting point or a hint that will get me going.  
Thanks in advance!
 A: We define irreducible ideal an ideal such that: if $I$ = $I_1 \cap I_2$, then $I$ = $I_1$ or $I$ = $I_2$.
To prove that an irreducible monomial ideal $I$ has a basis of pure powers of variables, you may use the following:
Lemma: Let $I$ be a monomial ideal in $K[X_1, ... , X_n]$ such that $I$ = $(m_1, ... , m_k, uv)$ with $u, v$ coprime, then $I$ = $(I, u) \cap (I, v)$ = $(m_1, ... , m_k, u) \cap (m_1, ... , m_k, v)$.
You can prove it by double inclusion.
Thus, if $I$ = $(m_1, ... ,m_k)$ had a generating monomial $m_i$ = $n_1n_2$ such that $n_1$ and $n_2$ have no common variables, then you could write $I$ as $(I, n_1) \cap (I, n_2)$ which is an intersection of ideals which strictly contain $I$, in contradiction whith the hypothesis of $I$ irreducible.
To prove the other implication suppose  $I$ = $I_1 \cap I_2$ (such that $I$ is not irreducible) and take two monomials $n_1 \in I_1 - I$ and $n_2 \in I_2 - I$. Then, since $I_1I_2 \subset I$, $n_1n_2 \in I$. To end the proof you have to prove that the only variables which compare in $n_i$ are the same of the basis of $I$ and that the $lcm(n_1, n_2)$ is divided by some $m_i$ in the basis, in contra, thus $n_1$ or $n_2$ is in $I$ (contradiction).
However in this second part there's just the idea of the proof but it needs to be formalized.
