What the study found
The study shows that for every singular cardinal, there is a valuation domain that is strictly (< aleph_alpha)-noetherian. For every regular cardinal, there is a valuation domain that is strictly (< aleph_alpha+)-noetherian.
Why the authors say this matters
The authors state that this gives a positive answer to a problem posed by Mazari-Armida under certain set theory assumption. In the paper's terms, this addresses the existence of rings with a specific kind of noetherian property at given cardinal sizes.
What the researchers tested
The article is a mathematical note about rings, specifically valuation domains. It examines whether, for each singular or regular cardinal, such domains exist with the stated strictly parameterized noetherian properties.
What worked and what didn't
The abstract says the existence result holds for every singular cardinal and for every regular cardinal, with the corresponding versions of strict (< aleph_alpha) and strict (< aleph_alpha+)-noetherianity. No unsuccessful cases or counterexamples are described in the abstract.
What to keep in mind
The abstract mentions that the positive answer is obtained under a certain set theory assumption, but it does not specify that assumption here. It also does not provide the proof details, examples, or any limitations beyond the stated scope.
Key points
- The paper proves the existence of valuation domains with strict cardinal-based noetherian properties.
- For every singular cardinal, the authors report a strictly (< aleph_alpha)-noetherian valuation domain.
- For every regular cardinal, the authors report a strictly (< aleph_alpha+)-noetherian valuation domain.
- The authors say this answers a problem posed by Mazari-Armida under a set theory assumption.
- The abstract does not describe the specific assumption or the proof details.
Disclosure
- Research title:
- Existence of strictly parameterized noetherian valuation domains shown
- Authors:
- Xiaolei Zhang
- Institutions:
- Tianshui Normal University
- Publication date:
- 2026-04-20
- OpenAlex record:
- View
Get the weekly research newsletter
Stay current with scholarly research without reading academic papers — one filtered digest, every Friday.