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Strong and Explicit Forms of the One-Sided Nullstellensatz over Division Rings
Abstract
We establish strong and explicit forms of a one-sided Nullstellensatz for polynomial rings D[x1, . . . , xn] over arbitrary division rings D with central indeterminates. To this end, we employ the theory of integral dependence over left ideals and the notion of G-left ideals, which plays a role analogous to that of G-ideals in the classical theory. We prove that the Jacobson radical of any left ideal of D[x1, . . . , xn] coincides with the intersection of all G-left ideals containing it. As a consequence, we obtain an explicit description of Jacobson radicals of left ideals of polynomial rings over division rings in terms of integral dependence.
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BibTeX
@misc{aryapoor2026strong,
title = {{Strong and Explicit Forms of the One-Sided Nullstellensatz over Division Rings}},
author = {Masood Aryapoor},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34278},
eprint = {2609.34278},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34278},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Strong and Explicit Forms of the One-Sided Nullstellensatz over Division Rings
AU - Masood Aryapoor
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34278
UR - https://arxiv.org/abs/2609.34278
AB - We establish strong and explicit forms of a one-sided Nullstellensatz for polynomial rings D[x1, . . . , xn] over arbitrary division rings D with central indeterminates. To this end, we employ the theory of integral dependence over left ideals and the notion of G-left ideals, which plays a role analogous to that of G-ideals in the classical theory. We prove that the Jacobson radical of any left ideal of D[x1, . . . , xn] coincides with the intersection of all G-left ideals containing it. As a consequence, we obtain an explicit description of Jacobson radicals of left ideals of polynomial rings over division rings in terms of integral dependence.
ER -