Let $B=\oplus_{n\in\mathbb Z} B_n$ be a graded ring (commutative with 1). We know that $B_0$ is a subring of $B$, so we have the inclusion $B_0\hookrightarrow B$.
My question is:
Is every prime ideal of $B_0$ the inverse image of some homogeneous prime ideal in $B$?
If $B_0$ is a Principal Ideal Domain, then it's true, so I was trying to find a counterexample in $\mathbb Z[t][x,y]$, but it's not so easy.