Graded differential polynomial rings

Abstract

We study differential polynomial rings $R[t;\delta ]$ over $\Gamma$-graded rings, where $\Gamma$ is an arbitrary group. We show that $R[t;\delta ]$ admits a $\Gamma$-grading compatible with that of $R$ if and only if $\delta$ is a $\gamma$-derivation for some $\gamma \in C_{\Gamma }({\Gamma }_R)$, and that this grading is unique once $°(t)=\gamma$ is fixed; if $\delta \ne 0$, then $\gamma$ is itself uniquely determined by $\delta$. We characterize the resulting graded ring by a universal property. We prove a characteristic-free center criterion for gr-simplicity whenever $Z(R[t;\delta ])$ is a graded subring; in characteristic zero, gr-simplicity is equivalent to $\delta$-gr-simplicity of $R$ and $\gamma$-outerness of $\delta$, extending Jordan's simplicity criterion to the graded setting. We further show that $R[t;\delta ]$ is gr-prime if and only if $R$ is $\delta$-gr-prime, and that gr-Noetherianity of $R$ passes to $R[t;\delta ]$, recovering a graded Hilbert basis theorem as a special case. When $\Gamma$ is abelian, gr-simplicity and gr-primality are shown to be invariants of homogeneous graded Morita equivalence, and every ring homogeneously graded equivalent to $R[t;\delta ]$ via a compatible idempotent is again a graded differential polynomial ring.