We have the following theorem:
Let $R$ be a graded ring s.t. $R_0$ is a local ring, in which residue field is infinite. Let $I$ be an ideal generated by homogeneous elements of degree $s$. Let $J_1,...,J_n$ be ideals of $R$ s.t. none of them contains $I$. Then we can find a homogeneous element $x\in I$ of degree $s$ s.t. $x\notin J_i \ \forall i$.
My question is, can we find a counterexample that the theorem is wrong if the residue field is not infinite? Thank you.