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.
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