  Let $H^2(\mathbb{D}^n)$ denote the Hardy space over the polydisc
$\mathbb{D}^n$, $n \geq 2$. A closed subspace $\mathcal{Q} \subseteq
H^2(\mathbb{D}^n)$ is called Beurling quotient module if there exists an inner
function $\theta \in H^\infty(\mathbb{D}^n)$ such that $\mathcal{Q} =
H^2(\mathbb{D}^n) /\theta H^2(\mathbb{D}^n)$. We present a complete
characterization of Beurling quotient modules of $H^2(\mathbb{D}^n)$: Let
$\mathcal{Q} \subseteq H^2(\mathbb{D}^n)$ be a closed subspace, and let
$C_{z_i} = P_{\mathcal{Q}} M_{z_i}|_{\mathcal{Q}}$, $i=1, \ldots, n$. Then
$\mathcal{Q}$ is a Beurling quotient module if and only if \[ (I_{\mathcal{Q}}
- C_{z_i}^* C_{z_i}) (I_{\mathcal{Q}} - C_{z_j}^* C_{z_j}) = 0 \qquad (i \neq
j). \] We present two applications: first, we obtain a dilation theorem for
Brehmer $n$-tuples of commuting contractions, and, second, we relate joint
invariant subspaces with factorizations of inner functions. All results work
equally well for general vector-valued Hardy spaces.
