Graded Differential Polynomial Rings

ABSTRACT We study differential polynomial rings over ‐graded rings, where is an arbitrary group. We show that admits a ‐grading compatible with that of if and only if is a ‐derivation for some , and that this grading is unique once is fixed; if , then is itself uniquely determined by . We characterize the resulting graded ring by a universal property. We prove a characteristic‐free center criterion for gr‐simplicity whenever is a graded subring; in characteristic zero, gr‐simplicity is equivalent to ‐gr‐simplicity of and ‐outerness of , extending Jordan's simplicity criterion to the graded setting. We further show that is gr‐prime if and only if is ‐gr‐prime, and that gr‐Noetherianity of passes to , recovering a graded Hilbert basis theorem as a special case. When is abelian, gr‐simplicity and gr‐primality are shown to be invariants of homogeneous graded Morita equivalence, and every ring homogeneously graded equivalent to via a compatible idempotent is again a graded differential polynomial ring.