Graded differential polynomial rings

Let $R$ be a $\mathrm{\Gamma }$-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 $\mathrm{°}(t)$ 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 Öinert–Silvestrov criterion when $\mathrm{\Gamma }$ 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.