# Irreducible homogeneous ideals

I have the following question:

Let $I$ be a homogeneous ideal. Is it true that $I$ is irreducible if and only if it can't be written as the intersection of two homogeneous ideals?

So, is it sufficient checking the irreducibility on homogeneous ideals? If it's true, how can I prove this result?

• What is the definition of an irreducible ideal ? – Cantlog Mar 17 '14 at 19:36
• @Cantlog An ideal which is not the intersection of two proper ideals properly containing it. – Alex Becker Mar 31 '14 at 6:23
• @user26857 This question is too old for moderators to migrate - but you could always just ask it on MO (and link back to here so people don't get upset about cross-posting). – Alex Becker Apr 2 '14 at 16:49
• Asked now on MO, too: mathoverflow.net/questions/162739/… – user26857 Apr 10 '14 at 21:11
• A proof posted in here: mathoverflow.net/questions/162739/… See [here][1] for a related question. [1]: mathoverflow.net/questions/215768/… – Pham Hung Quy Sep 1 '15 at 3:01