in Tom Marley's note on Graded Ring and Module, there's a theorem stated "Let $R$ be a nonnegatively graded Noetherian ring, $R_0$ is a local and Artinian ring. Let $M$ be an maximal homogeneous ideal and $d=dim R=ht(M)$. Then there exists homogeneous elements $w_1,...,w_d\in R_+$ s.t. $M=\sqrt{(w_1,...,w_d)}$."
And there's a remark below, saying that this is not true if $R_0$ is not Artinian. Consider $R=\mathbb{Z}_{(2)}[x]/(2x)$, then $dim R=1$ and $M=(2,x)R$, but there isn't a homogeneous element $w\in R$ s.t. $M=\sqrt{(w)}$.
I can hardly imagine what $R$ is so I have difficulty understanding the counterexample. Why is $dimR=1$ and why is $M\neq \sqrt{(w)}$? Thank you