is proved analogously to the classical case.

\[pr:TPP super toric q\] Let $({\bf x},B,\Lambda)$ be a quantum seed, $rank(\Lambda)=n$ and $\II$ a non-zero two-sided ideal. Then the ideal $\II$ contains a monomial in $x^m\in \CC_\Lambda[{\bf x}]$.

We do not have a quantum version of Proposition \[th:intersection\] because we do not know whether prime ideals in quantum cluster algebras are completely prime.

Poisson ideals in acyclic cluster algebras {#se:Poisson and symp geom}
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In this section we recall results from our previous paper [@ZW; @tpc]. As the proofs are rather short, we shall include them for convenience. Recall e.g. from [@BFZ] that acyclic cluster algebras associated with an acyclic quiver and with trivial coefficients correspond, up to a reordering of the variables of the acyclic seed, to cluster algebras defined by a seed $({\bf x},B)$ where $B$ is a skew-symmetric $n\times n$-matrix with $b_{ij}>0$ if $i<j$.

Berenstein, Fomin and Zelevinsky proved in [@BFZ] that such a cluster algebra $\AA$ is equal to both its lower and upper bounds. Thus, it is Noetherian and, if $B$ has full rank, a Poisson algebra with the Poisson brackets given by compatible pairs $(B,\Lambda)$ with $\Lambda=\mu B^{-1}$ for certain $\mu\in \ZZ$ (see Example \[ex:acyclic\]). In 