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We study a division problem in the Hardy classes $H^p\left(\right)$ of the unit ball of ${ℂ}^2$ which generalizes the $H^p$ corona problem, the generators being allowed to have common zeros. MPrecisely, if S is a subset of , we study conditions on a ${ℂ}^k$-valued bounded Mholomorphic function B, with $B_{|S}=0$, in order that for 1 ≤ p < ∞ and any function $f\in H^p\left(\right)$ with $f_{|S}=0$ there is a ${ℂ}^k$-valued $H^p\left(\right)$ holomorphic function F with f = B·F, i.e. the module generated by the components of B in the Hardy class $H^p\left(\right)$ is the entire module $M_S:=f\in H^p\left(\right):f_{|S}=0$. As a special case,...