Graded differential polynomial rings
Abstract
We study differential polynomial rings R[t;δ] over Γ-graded rings, where Γ is an arbitrary group. We show that R[t;δ] admits a Γ-grading compatible with that of R if and only if δ is a γ-derivation for some γ∈ CΓ(ΓR), and that this grading is unique once °(t)=γ is fixed; if δ≠0, then γ is itself uniquely determined by δ. We characterize the resulting graded ring by a universal property. We prove a characteristic-free center criterion for gr-simplicity whenever $Z(R[t;δ])$ is a graded subring; in characteristic zero, gr-simplicity is equivalent to δ-gr-simplicity of R and γ-outerness of δ, extending Jordan's simplicity criterion to the graded setting. We further show that R[t;δ] is gr-prime if and only if R is δ-gr-prime, and that gr-Noetherianity of R passes to R[t;δ], recovering a graded Hilbert basis theorem as a special case. When Γ 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;δ] via a compatible idempotent is again a graded differential polynomial ring.
Versions
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BibTeX
@misc{mohamed2026graded,
title = {{Graded differential polynomial rings}},
author = {Yassine Ait Mohamed},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.23352},
eprint = {2606.23352},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.23352},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Graded differential polynomial rings
AU - Yassine Ait Mohamed
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2606.23352
UR - https://arxiv.org/abs/2606.23352
AB - We study differential polynomial rings R[t;δ] over Γ-graded rings, where Γ is an arbitrary group. We show that R[t;δ] admits a Γ-grading compatible with that of R if and only if δ is a γ-derivation for some γ∈ C_Γ(Γ_R), and that this grading is unique once °(t)=γ is fixed; if δ≠0, then γ is itself uniquely determined by δ. We characterize the resulting graded ring by a universal property. We prove a characteristic-free center criterion for gr-simplicity whenever $Z(R[t;δ])$ is a graded subring; in characteristic zero, gr-simplicity is equivalent to δ-gr-simplicity of R and γ-outerness of δ, extending Jordan's simplicity criterion to the graded setting. We further show that R[t;δ] is gr-prime if and only if R is δ-gr-prime, and that gr-Noetherianity of R passes to R[t;δ], recovering a graded Hilbert basis theorem as a special case. When Γ 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;δ] via a compatible idempotent is again a graded differential polynomial ring.
ER -