Rings whose subrings are all Noetherian or Artinian

Nathan Blacher

arxiv: 2402.13633 · v2 · submitted 2024-02-21 · 🧮 math.RA

Rings whose subrings are all Noetherian or Artinian

Nathan Blacher This is my paper

Pith reviewed 2026-05-24 04:16 UTC · model grok-4.3

classification 🧮 math.RA

keywords noncommutative ringsNoetherian ringsArtinian ringssubringstrivial extensionsPrüfer p-groupchain conditionsPI rings

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The pith

If every proper subring of a ring is right Noetherian, then the ring itself is right Noetherian unless it is the trivial extension of Z by the Prüfer p-group for a prime p.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper classifies noncommutative rings in which every proper subring satisfies a given chain condition on one-sided ideals. When all proper subrings are right Noetherian, the whole ring must be right Noetherian except for one explicit construction; when all proper subrings are right Artinian, the ring is right Artinian or simply the integers. These statements extend earlier commutative results, though the Artinian case generalizes only the absolute version and the Noetherian case requires no extra hypotheses. A sympathetic reader cares because the results show that the chain-condition property on proper subrings propagates to the ring with only isolated exceptions, limiting how noncommutative rings can behave differently from commutative ones.

Core claim

If every proper subring of a ring R is right Noetherian, then R is either right Noetherian or the trivial extension of Z by the Prüfer p-group for a prime p. If every proper subring of R is right Artinian, then R is either right Artinian or Z. The Artinian result generalizes only the absolute commutative case, while the full commutative Artinian statement is recovered for PI rings when only certain subrings need be Artinian.

What carries the argument

The trivial extension of Z by the Prüfer p-group, which is the unique ring whose proper subrings are all right Noetherian but which itself fails to be right Noetherian.

If this is right

The only rings that can fail to be right Noetherian while keeping all proper subrings right Noetherian are the listed trivial extensions.
The only rings that can fail to be right Artinian while keeping all proper subrings right Artinian are the integers themselves.
For PI rings the Artinian conclusion holds even when the Artinian condition is imposed only on a designated family of subrings rather than all of them.
The commutative classification of Gilmer and Heinzer survives passage to the noncommutative setting without additional counterexamples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The scarcity of exceptions suggests that one-sided chain conditions on subrings are rigid enough that noncommutative constructions rarely produce fresh pathologies.
One could check whether similar statements hold for other ascending-chain conditions such as Goldie or Noetherian on both sides by examining the same trivial-extension family.
Matrix rings over the exceptional examples might be tested to see whether the subring condition survives or immediately forces the full ring to satisfy the chain condition.

Load-bearing premise

The way right ideals sit inside noncommutative trivial extensions does not create new rings whose proper subrings obey the chain condition while the ring itself does not.

What would settle it

A concrete ring that is neither right Noetherian nor the trivial extension of Z by any Prüfer p-group, yet has every proper subring right Noetherian.

read the original abstract

We study noncommutative rings whose proper subrings all satisfy the same chain condition. We show that if every proper subring of a ring $R$ is right Noetherian, then $R$ is either right Noetherian or the trivial extension of $\mathbb{Z}$ by the Pr\"ufer $p$-group for a prime $p$. We also prove that if every proper subring of $R$ is right Artinian, then $R$ is either right Artinian or $\mathbb{Z}$. For commutative rings, both results were proved by Gilmer and Heinzer in 1992. Our result for right Artinian subrings only generalises the absolute case of their commutative result. We generalise the full result (when only certain subrings are right Artinian) in the context of PI rings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends the 1992 Gilmer-Heinzer commutative classification to noncommutative rings for the Noetherian case fully and the Artinian case in absolute and PI settings.

read the letter

The paper classifies noncommutative rings where every proper subring is right Noetherian or right Artinian. It carries over the Gilmer-Heinzer results with the right Noetherian theorem stating that R is right Noetherian or the trivial extension of Z by the Prüfer p-group, and the Artinian theorem stating that R is right Artinian or Z, plus a PI-ring version that recovers the full conditional commutative result. These are the new pieces relative to the cited work. The statements are direct, use standard definitions, and name the exceptional cases explicitly without circularity or invented entities. The adaptation to noncommutative rings appears to go through cleanly on the stated claims, with no obvious new counterexamples or hidden assumptions in the setup. The main limitation is that the Artinian result stays partial outside the PI case, which the authors flag themselves. This is a real scoping choice rather than a flaw in execution, but it leaves the noncommutative version less complete than the original commutative one. The work targets ring theorists who care about chain conditions on subrings. A reader already working on noncommutative Noetherian or Artinian rings would get concrete reference classifications from it. The evidence in the abstract and stress-test note supports sending it to peer review so the proofs can be checked in detail.

Referee Report

0 major / 3 minor

Summary. The paper extends the 1992 Gilmer-Heinzer classification of commutative rings in which every proper subring satisfies a chain condition to the noncommutative setting. It proves that if every proper subring of a ring R is right Noetherian, then R is right Noetherian or the trivial extension of Z by the Prüfer p-group for a prime p. If every proper subring is right Artinian, then R is right Artinian or Z; the full conditional version is obtained when R is a PI-ring.

Significance. If the stated theorems hold, the work supplies explicit, noncommutative analogues of the Gilmer-Heinzer results, relying on standard constructions such as trivial extensions and the cited commutative classification. The absolute Noetherian case and the absolute Artinian case are fully treated, with an additional PI-ring extension for the conditional Artinian statement; these are concrete classification theorems rather than existence results.

minor comments (3)

The abstract and introduction should explicitly state the precise definition of the trivial extension Z ⋉ Z(p^∞) used in the Noetherian theorem, including the module action, to avoid ambiguity for readers unfamiliar with the construction.
In the discussion of the Artinian result, the manuscript notes that only the absolute case is generalized; a brief remark on the obstruction preventing the full conditional generalization outside the PI setting would improve clarity.
The paper cites Gilmer-Heinzer (1992); adding a short comparison paragraph highlighting which steps of their commutative argument adapt directly and which require new noncommutative arguments would strengthen the exposition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of the manuscript and the recommendation of minor revision. The report lists no specific major comments under the MAJOR COMMENTS heading.

Circularity Check

0 steps flagged

No significant circularity; classification is self-contained

full rationale

The paper presents direct classification theorems for noncommutative rings based on standard definitions of Noetherian/Artinian rings and subrings, extending the external 1992 Gilmer-Heinzer commutative results via independent arguments. No load-bearing steps reduce by construction to fitted inputs, self-definitions, or self-citation chains; the commutative citations are external and the noncommutative extensions use explicit ring constructions (e.g., trivial extensions) without circular reduction. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard definitions from ring theory without introducing new free parameters or invented entities. All background is drawn from prior literature on Noetherian/Artinian rings and specific constructions like trivial extensions.

axioms (2)

standard math Standard definitions of right Noetherian and right Artinian rings (ascending/descending chain conditions on right ideals)
Invoked throughout to state the hypotheses and conclusions.
standard math Properties of the trivial extension construction and the Prüfer p-group
Used to identify the exceptional non-Noetherian example.

pith-pipeline@v0.9.0 · 5660 in / 1463 out tokens · 61336 ms · 2026-05-24T04:16:18.042245+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

Gilmer and W

R. Gilmer and W. Heinzer. Noetherian pairs and hereditar ily noetherian rings. Archiv der Mathematik , 41:131–138, 1983

work page 1983
[2]

Gilmer and W

R. Gilmer and W. Heinzer. An application of jonsson modul es to some questions con- cerning proper subrings. Mathematica Scandinavica, 70:34–42, 1992

work page 1992
[3]

Gilmer and M

R. Gilmer and M. O’Malley. Non-noetherian rings for whic h each proper subring is noetherian. Mathematica Scandinavica, 31(1):118–122, 1972

work page 1972
[4]

T. Y. Lam. Lectures on Modules and Rings . Graduate Texts in Mathematics. Springer New York, NY, 1998

work page 1998
[5]

J. C. McConnell and J. C. Robson. Noncommutative Noetherian rings . Chichester : Wiley, 1987

work page 1987
[6]

L. H. Rowen. Polynomial Identities in Ring Theory . Academic Press, 1980. School of Mathematics and Statistics University of Sheﬃeld Hicks Building Sheﬃeld S3 7RH United Kingdom email: nblacher1@sheﬃeld.ac.uk 9

work page 1980

[1] [1]

Gilmer and W

R. Gilmer and W. Heinzer. Noetherian pairs and hereditar ily noetherian rings. Archiv der Mathematik , 41:131–138, 1983

work page 1983

[2] [2]

Gilmer and W

R. Gilmer and W. Heinzer. An application of jonsson modul es to some questions con- cerning proper subrings. Mathematica Scandinavica, 70:34–42, 1992

work page 1992

[3] [3]

Gilmer and M

R. Gilmer and M. O’Malley. Non-noetherian rings for whic h each proper subring is noetherian. Mathematica Scandinavica, 31(1):118–122, 1972

work page 1972

[4] [4]

T. Y. Lam. Lectures on Modules and Rings . Graduate Texts in Mathematics. Springer New York, NY, 1998

work page 1998

[5] [5]

J. C. McConnell and J. C. Robson. Noncommutative Noetherian rings . Chichester : Wiley, 1987

work page 1987

[6] [6]

L. H. Rowen. Polynomial Identities in Ring Theory . Academic Press, 1980. School of Mathematics and Statistics University of Sheﬃeld Hicks Building Sheﬃeld S3 7RH United Kingdom email: nblacher1@sheﬃeld.ac.uk 9

work page 1980