A Basis Theorem for Rings with Commuting Operators in Characteristic Zero

TL;DR

The paper develops a unified framework for basis theorems in polynomial rings equipped with commuting generalized Hasse–Schmidt operators, via the notion of $\\mathcal{D}^*$-rings. It proves that if $R$ is a $\\mathbb{Q}$-algebra with ACC on perfect $\\mathcal{D}$-ideals, then the $\\mathcal{D}^*$-polynomial ring $R\{\bar{x}\\}_{\\mathcal{D}^*}$ inherits ACC on perfect $\\mathcal{D}^*$-ideals, thereby generalizing both Kolchin’s differential basis theorem and Cohn’s difference–differential basis theorem. The approach relies on reducing to ranked bases, establishing a division algorithm via the $\\mathcal{D}^*$-reduction lemma, and leveraging characteristic sets within a perfect conservative system of ideals. This framework not only recovers known results in characteristic zero but also enables new cases and connections among differential, difference, and their hybrid theories. The results have implications for the structural analysis of ideals in $\\mathcal{D}^*$-polynomial rings and for transfer of Noetherian properties across operator-augmented polynomial rings.

Abstract

Motivated by the differential basis theorem of Kolchin and the difference-differential basis theorem of Cohn, in this paper we present a basis theorem for polynomial rings equipped with commuting generalised Hasse-Schmidt operators (in the sense of Moosa and Scanlon). We recover Kolchin and Cohn's results as special cases of our main theorem.

Theorems & Definitions (59)

• Theorem 1.1: $\mathcal{D}$-Basis Theorem
• Remark 2.1
• Example 2.2
• Definition 2.3: $\mathcal{D}$-ideal
• Definition 2.4
• Definition 2.5: $\mathcal{D}^*$-ring
• Lemma 2.6
• proof
• Definition 2.7: $\mathcal{D}^*$-polynomial ring
• Lemma 2.8