Prime Avoidance Theorem - graded version:
Let $R$ be a graded ring and $I$ is a homogeneous ideal generated by homogeneous elements of positive degree. Let $J_1,...,J_n$ be homogeneous prime ideals such that none of them contains $I$. Then we can find $x\in I$ such that $x$ is homogeneous and $x \notin P_i\ \forall i$.
Can we find a counterexample for this theorem when $I$ isn't generated by homogeneous elements of positive degree?