Is every minimal set of generators for a homogeneous ideal composed by homogeneous elements?

Question

An ideal $\mathfrak a$ of a graded ring $A$ is said to be homogeneous if I can find a set of homogeneous generators for $\mathfrak a$. Is it true that every minimal set of generators for a homogeneous ideal $\mathfrak a$ is composed by homogeneous elements?

Thanks in advance

Answer 1

The answer is no. Consider the following homogenous ideal of $\mathbb{C}[x,y,z]$ :

$$ I = (x^2+y^2+z^2, xyz, z^5).$$

This is also equal to $ I = (x^2+y^2+z^2, xyz+z^5, xyz- z^5).$