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PreprintCC BY-NC-SA 4.0arXiv

Integral Dependence over One-Sided Ideals, Part II: G-Left Ideals and the Köthe Problem

  • Masood Aryapoor
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Abstract

We study integral dependence over left ideals using G-left ideals, namely left ideals maximal with respect to the failure of integrality of a fixed element. We establish several characterizations of G-left ideals, and show that their intersection is precisely the upper nilradical. Motivated by the Köthe conjecture, we introduce the class of left KI-rings as rings in which sums of left ideals integral over a fixed left ideal remain integral over that ideal. It is shown that left artinian rings, PI-rings, algebraic algebras, and algebras of dimension smaller than the cardinality of the ground field are left KI-rings.

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Cite

BibTeXRIS
BibTeX
@misc{aryapoor2026integral,
  title = {{Integral Dependence over One-Sided Ideals, Part II: G-Left Ideals and the Köthe Problem}},
  author = {Masood Aryapoor},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26909},
  eprint = {2609.26909},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26909},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Integral Dependence over One-Sided Ideals, Part II: G-Left Ideals and the Köthe Problem
AU  - Masood Aryapoor
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26909
UR  - https://arxiv.org/abs/2609.26909
AB  - We study integral dependence over left ideals using G-left ideals, namely left ideals maximal with respect to the failure of integrality of a fixed element. We establish several characterizations of G-left ideals, and show that their intersection is precisely the upper nilradical. Motivated by the Köthe conjecture, we introduce the class of left KI-rings as rings in which sums of left ideals integral over a fixed left ideal remain integral over that ideal. It is shown that left artinian rings, PI-rings, algebraic algebras, and algebras of dimension smaller than the cardinality of the ground field are left KI-rings.
ER  -