What the study found
The study shows that for every singular cardinal, there is a valuation domain that is strictly (<α)-noetherian. For every regular cardinal, it shows a valuation domain that is strictly (<α⁺)-noetherian.
Why the authors say this matters
The authors say this gives a positive answer to a problem posed by Mazari–Armida under certain set theory assumptions. In the paper's terms, this addresses the existence of parameterized noetherian rings for the cardinals considered.
What the researchers tested
The note studies a class of rings called left strictly (<α)-noetherian rings, meaning every ideal is generated by fewer than α elements for the least such cardinal. The author examines valuation domains and the cases of singular and regular cardinals.
What worked and what didn't
The paper reports existence results in both cases: singular cardinals correspond to strictly (<α)-noetherian valuation domains, while regular cardinals correspond to strictly (<α⁺)-noetherian valuation domains. The abstract does not describe any negative results or cases where the construction fails.
What to keep in mind
The abstract says the result holds under certain set theory assumptions, but it does not specify them here. It also does not provide proof details or additional limitations in the available summary.
Key points
- The paper proves existence of valuation domains with parameterized noetherian properties.
- For singular cardinals, the domains are strictly (<α)-noetherian.
- For regular cardinals, the domains are strictly (<α⁺)-noetherian.
- The authors say this answers a problem posed by Mazari–Armida under certain set theory assumptions.
- The abstract does not list proof details or further limitations.
Disclosure
- Research title:
- Parameterized noetherian rings exist for each cardinal case
- Authors:
- Xiaolei Zhang
- Institutions:
- Tianshui Normal University
- Publication date:
- 2026-04-20
- OpenAlex record:
- View
Get the weekly research newsletter
Stay current with peer-reviewed research without reading academic papers — one filtered digest, every Friday.