Graded differential polynomial rings

Abstract

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