We study the closed ideal in the Beurling algebras $\aA^{+}_{\alpha,\beta}$ of holomorphic function $f$ in the bidisc such that $$\sum\limits_{n,m\geq 0} |\widehat{f}(n,m)|(1+n)^{\alpha}(1+m)^\beta<\infty.$$ We characterize the functions $f\in \aA^+_{\alpha,\beta}$, under a restriction on their zero sets, such that the closed ideal generated by $f$ coincides with the ideal of all functions vanishing on the zero set of $f$.