Ideaux fermes d'algebres de Beurling analytiques sur le bidisque

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.