Nope it's ok, I found what I wanted. Thank you very much!

Thank you very much. I think I will need some more insight about this. Do you have any references?

I don't quite understand where this goes. With what you said, we prove that our minimal set of homogeneous elements generates the ring. But I don't get why they are algebraically independent.

I don't really understand why a minimal set of homogeneous generators for the ideal would be a minimal set of generators of the ring. Of course, it is a minimal set of homogeneous generators for the ring, but is it a minimal set of generators for the ring?