One knows that graded ideals in polynomial rings over a field are primary iff they are graded-primary. What about the irreducible ideals?

Let $I$ be a graded ideal in a polynomial ring over a field. Is it true that $I$ is irreducible if and only if it can't be written as the intersection of two graded ideals (properly containing $I$)?

If the answer is positive in this case, then what's going on if change the ring by an arbitrary $\Bbb N$ (or $\Bbb Z$)-graded ring?

One knows that graded ideals in polynomial rings over a field are primary iff they are graded-primary. What about the irreducible ideals?

Let $I$ be a graded ideal in a polynomial ring over a field. Is it true that $I$ is irreducible if and only if it can't be written as an intersection of two graded ideals (properly containing $I$)?

If the answer is positive in this case, then what's going on if change the ring by an arbitrary $\Bbb N$ (or $\Bbb Z$)-graded ring?