# Relation about prime ideals in $B$ and invariant subring $B^G$

Suppose $B$ is a commutative ring, $G$ is a finite group acting on $B$, $A=B^G$ is the invariant subring. Suppose $P$ is a prime ideal in $A$, $Q_1,...,Q_s$ are all the prime ideals in $B$ such that $P=A\cap Q_i$. And assume $s=|G|$.

Does it hold that $Q_1\cap\cdots\cap Q_s=PB$?