Homogeneus primes in a graded ring

Question

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.

Answer 1

Let $A$ be a $\mathbb Z$-graded commutative ring, and $\mathfrak p_0$ be a prime ideal of $A_0$. Then there is a graded prime ideal $P$ of $A$ such that $P_0=\mathfrak p_0$. (Here $P_0$ denotes degree zero component of $P$, that is, $P\cap A_0$.)

For each $n\in\mathbb Z$ set $P_n=\{a\in A_n:aA\cap A_0\subseteq\mathfrak p_0\}$. Now set $P=\bigoplus_{n\in\mathbb Z}P_n$. Then $P$ is a prime ideal of $A$ and $P_0=\mathfrak p_0$.