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arXiv · 2606.23352

Graded differential polynomial rings

Yassine Ait Mohamed
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Abstract

Let $R$ be a $\Gamma$-graded ring and $\delta$ a derivation of $R$. We determine exactly when the differential polynomial ring $R[t;\delta]$ admits a grading compatible with that of $R$: this happens if and only if $\delta$ is a $\gamma$-derivation for some $\gamma$ in the centralizer of the support, in which case the grading is explicit and unique once $\deg(t)$ is fixed. Over an arbitrary group, we establish graded analogues of the classical simplicity, primeness, and Noetherianity theorems; in characteristic zero, $R[t;\delta]$ is gr-simple if and only if $R$ is $\delta$-gr-simple and $\delta$ is $\gamma$-outer, and in arbitrary characteristic we obtain a graded \"{O}inert--Silvestrov criterion when $\Gamma$ is orderable and the nonzero homogeneous elements of $R[t;\delta]$ are regular. Finally, we show that the differential polynomial structure is invariant under homogeneous graded equivalence.

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Yassine Ait Mohamed. 2026-06-22. Graded differential polynomial rings. https://arxiv.org/abs/2606.23352

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