Ideals of homomorphic images of the enveloping algebra of the Witt algebra

Let $W_{\ge -1}=\mathbb{C}[t]\mathrm{\partial }$ and $W=\mathbb{C}[t,t^{-1}]\mathrm{\partial }$ be the Witt algebra of algebraic vector fields on $\mathbb{C}$ and ${\mathbb{C}}^{\ast }$ respectively. In this paper, we make significant progress toward the open conjecture that the enveloping algebras $\mathrm{U}(W_{\ge -1})$ and $\mathrm{U}(W)$ satisfy the ascending chain condition (ACC) on two-sided ideals. We show that all homomorphic images of $\mathrm{U}(W_{\ge -1})$ and $\mathrm{U}(W)$ under the family of orbit homomorphisms'' of arbitrary Gelfand-Kirillov dimension satisfy ACC on ideals. These orbit homomorphisms were the key ingredient allowing us to lift’’ the Dixmier map from finite-dimensional solvable settings to infinite-dimensional settings of the Witt and Virasoro algebras in our recent work [Pham, 2025, arXiv:2504.14670]. As a result, we completely classify the prime and primitive spectra of these homomorphic images. As these images approximate $\mathrm{U}(W_{\ge -1})$ better as their GK-dimension increases, this classification sheds new light on the two-sided and prime ideal structures of $\mathrm{U}(W_{\ge -1})$. Finally, we discuss several applications of our results to the Dixmier map for $W_{\ge -1}$.