How can one show that every nonzero element $x$ of the ring $\mathbb{Z}[\sqrt{35}]$ is contained in finitely many ideals? It is obvious in case of $x$ being invertible, but a general case is out of my sight. Is something special about the number $35$ (except it is composite)? The ring is not UFD ($35=5\cdot 7=\sqrt{35}\cdot\sqrt{35}$), and so neither it is PID, thus the standard factorization argument does not work here. However, this ring is Noetherian -- maybe it would be helpful somehow?
I will appreciate any hints. TIA.