order for $B$ to have full rank we have to assume that $n=2k$ is even. Let $P_i=m_i^+ +m_i^-$ where $m_i^+$ and $m_i^-$ denote the monomial terms in the exchange polynomial. Then $\{y_i,x_i\}=\mu_1m_i^+ +\mu_2m_i^-$ for some $\mu_1,\mu_2\in \ZZ$. We, additionally, want to require that $\mu_1\ne \mu_2$. To assure this, we assume that $$\label{eq:poisson gen}
\sum_{j=1}^n (b^{-1})_{ij}\left(max(b_{ij},0)+min(b_{ij},0)\right)\ne 0$$ for all $i\in [1,n]$. We have the following result.

[@ZW; @tpc] \[th:acyclic\] Let $\AA$ be an acyclic cluster algebra over $\CC$ with trivial coefficients of even rank $n=2k$, given by a seed $(x_1,\ldots, x_{n}, B)$ where $B$ is a skew-symmetric $n\times n$-integer matrix satisfying $b_{ij}>0$ if $i<j$ and suppose that $B$ and $B^{-1}$ satisfy Equation \[eq:poisson gen\] for each $i\in[1,n]$. Then, the Poisson cluster algebra defined by a compatible pair $(B,\Lambda)$ where $\Lambda=\mu B^{-1}$ with $0\ne \mu\in \ZZ$ contains no non-trivial Poisson prime ideals.

Suppose that there exists a non-trivial Poisson prime ideal $\II$. Then, $\II\cap {\bf x}$ is nonempty by Proposition \[pr:TPP super toric\], hence $\II\cap {\bf x}=\{x_{i_1},\ldots, x_{i_j}\}$ for some $1\le i_1\le i_2\le \ldots\le i_j\le 2k$. Additionally, observe that $P_{i_1}=m_{i_1}^++m_{i_1}^-$ has to be contained in $\II$, as well as $$\{y_{i_1},x_{i_1}\}=\mu_1m_{i_1}^+ +\mu_2m_{i_1}^-\ .$$ By our assumption, we have $\mu_1\ne \mu_2$, and therefore $m_{i_1}^-\in 