  We study the closed ideal in the Beurling algebras
$\mathcal{A}^{+}_{\alpha,\beta}$ of holomorphic function $f$ in the bidisc such
that $\sum_{n,m\geq 0}|\hat{f}(n,m)|(1+n)^{\alpha}(1+m)^\beta<+\infty$. We
determine the function $f\in\mathcal{A}^{+}_{\alpha,\beta}$ such that the
ideals generated by $f$ coincide with the ideal generated by their zeros set.
