Exercise $2.18$ in Eisenbud's algebra book asks to prove:
Suppose $R=\bigoplus_{n=-\infty}^\infty R_n$ is a $\Bbb Z$-graded ring such that any homogeneous prime ideal is zero. Prove $R_0$ is a field.
To do the first part I have tried to show that any maximal ideal $M$ of $R_0$ is zero, by somehow using $M$ to build a prime homogeneous ideal of $R$ containing $M$, but I can't see how.
