in qca}
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\[th:ideals in qclalg\] Let $\AA_q$ be a quantum cluster algebra with quantum seed $({\bf x}, B,\Lambda)$ satisfying Equation \[eq:poisson gen\]. Then, $\AA_q$ does not contain any non-trivial proper prime ideals.

Let $\II$ be a prime ideal in $\AA_q$. We obtain from Proposition \[pr:TPP super toric q\] that $\II$ contains a monomial $x^{\bf v}$ with ${\bf v}\in\ZZ_{\ge 0}^n$. It is easy to observe that we can choose ${\bf v}$ to be minimal with respect to the lexicographic order on $\ZZ^n_{\ge 0}$. Recall that the lexicographic ordering defines ${\bf u}<{\bf w}$ for ${\bf u},{\bf w}\in \ZZ^n$ if and only if there exists $i\in [1,n]$ such that $u_i,w_i$ and $u_j=w_j$ if $j>i$. Recall that this defines a total ordering on $\ZZ^n_{\ge0}$. There exists some $i\in[1,n]$ such that $v_i\ne 0$ and $v_k=0$ for all $k>i$, and we write $x^{\bf v}=x^{\bf v'} x_i^{v_i}$ with ${\bf v'}={\bf v}-v_i{\bf  e}_i$.

Recall that an ideal $\II$ in a non-commutative ring $R$ is prime if $arb\in \II$ for all $r\in R$ implies that $a\in \II$ or $b\in \II$. Now, since $\AA_q$ is acyclic, we know from [@bz-qclust Theorem 7.3 and 7.5] that it is isomorphic to its lower bound. Hence, employing the notation of Definition \[def:lower 