An Inductive Proof of Stafford's Theorem

Let \( K \) be an algebraically closed field of characteristic zero. The generalized Weyl algebra \( A_{n,f} \) is defined by generators \( x_1, x_2, \dots, x_n, y_1, \dots, y_n, z_1, \dots, z_n \) subject to certain commutation relations and additional structure determined by a collection of functions \( f = (f_1, \dots, f_n) \). We focus on the structure of left and right ideals in \( A_{n,f}(K) \), particularly proving that every left or right ideal can be generated by two elements. The proof is based on showing that if a left ideal can be generated by three elements, it can be reduced to two elements by applying the Noetherian property of the ring and an iterative reduction process. This result complements the simplicity of \( A_{n,f}(K) \), as established in prior work.