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PreprintCC BY-NC-ND 4.0arXiv

Integral Dependence over One-Sided Ideals, Part I: Foundations and Classical Applications

  • Masood Aryapoor
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Abstract

We introduce notions of integrality and full integrality over one-sided ideals, together with associated radical constructions, and develop a general framework for their study. As applications, we extend several classical results concerning Jacobson radicals to the one-sided setting. In particular, we prove that the Jacobson radical of a left ideal is integral over that ideal in left artinian rings and in algebraic algebras, obtaining a one-sided analogue of a theorem of Amitsur. We also show that Jacobson radicals of graded left ideals are graded and derive a corresponding extension of Amitsur's theorem on Jacobson radicals of polynomial rings.

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Cite

BibTeXRIS
BibTeX
@misc{aryapoor2026integral,
  title = {{Integral Dependence over One-Sided Ideals, Part I: Foundations and Classical Applications}},
  author = {Masood Aryapoor},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05935},
  eprint = {2609.05935},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05935},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Integral Dependence over One-Sided Ideals, Part I: Foundations and Classical Applications
AU  - Masood Aryapoor
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.05935
UR  - https://arxiv.org/abs/2609.05935
AB  - We introduce notions of integrality and full integrality over one-sided ideals, together with associated radical constructions, and develop a general framework for their study. As applications, we extend several classical results concerning Jacobson radicals to the one-sided setting. In particular, we prove that the Jacobson radical of a left ideal is integral over that ideal in left artinian rings and in algebraic algebras, obtaining a one-sided analogue of a theorem of Amitsur. We also show that Jacobson radicals of graded left ideals are graded and derive a corresponding extension of Amitsur's theorem on Jacobson radicals of polynomial rings.
ER  -