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arxiv: 1907.00946 · v2 · pith:EKTPZCHSnew · submitted 2019-07-01 · 🧮 math.RA

Uniserial Noetherian Centrally Essential Rings

Victor Markov , Askar Tuganbaev This is my paper

Pith reviewed 2026-05-25 11:03 UTC · model grok-4.3

classification 🧮 math.RA
keywords uniserial ringsNoetherian ringsArtinian ringscentrally essential ringsdiscrete valuation domainsring classification
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0 comments X

The pith

A right uniserial right Noetherian centrally essential ring is either a commutative discrete valuation domain or a two-sided Artinian uniserial ring.

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

The paper proves that right uniserial right Noetherian centrally essential rings are precisely the commutative discrete valuation domains together with the rings that are uniserial and Artinian on both sides. It supplies explicit non-commutative examples in the Artinian case. A sympathetic reader cares because the result gives a complete structural description that reduces an abstract combination of properties to two familiar classes of rings.

Core claim

A ring R is a right uniserial, right Noetherian centrally essential ring if and only if R is a commutative discrete valuation domain or a left and right Artinian, left and right uniserial ring. There also exist non-commutative uniserial Artinian centrally essential rings.

What carries the argument

The if-and-only-if classification that equates right uniserial right Noetherian centrally essential rings with commutative discrete valuation domains or two-sided Artinian uniserial rings.

If this is right

  • Any such ring that fails to be Artinian on the left must be a commutative discrete valuation domain.
  • Non-commutative examples are possible only inside the two-sided Artinian uniserial class.
  • The right Noetherian hypothesis forces the left Artinian property whenever the ring is non-commutative.

Where Pith is reading between the lines

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

  • The classification implies that the centrally essential condition imposes commutativity once the ring is Noetherian but not Artinian.
  • Similar equivalences might be checked for left Noetherian uniserial rings under the same central condition.
  • The non-commutative Artinian examples show that uniseriality alone does not force the center to be large enough without the Artinian hypothesis.

Load-bearing premise

The standard definitions of uniserial, Noetherian, Artinian, discrete valuation domain, and centrally essential are taken exactly as in the prior literature.

What would settle it

Construction of a ring that is right uniserial and right Noetherian, centrally essential, yet neither a commutative discrete valuation domain nor left and right Artinian uniserial.

read the original abstract

It is proved that a ring $R$ is a right uniserial, right Noetherian centrally essential ring if and only if $R$ is a commutative discrete valuation domain or a left and right Artinian, left and right uniserial ring. It is also proved that there exist non-commutative uniserial Artinian centrally essential rings. Victor Markov is supported by the Russian Foundation for Basic Research, project 17-01-00895-A. Askar Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove that a ring R is right uniserial, right Noetherian and centrally essential if and only if it is either a commutative discrete valuation domain or a left-and-right Artinian left-and-right uniserial ring; it also claims to establish the existence of non-commutative examples of uniserial Artinian centrally essential rings.

Significance. If the equivalence is correctly proved from standard ring-theoretic axioms and prior lemmas on uniserial and Artinian rings, the result supplies a clean classification that organizes the intersection of three well-studied classes (uniserial, Noetherian, centrally essential). The explicit construction of non-commutative examples would also be a concrete contribution.

major comments (1)
  1. [Abstract] The provided text consists solely of the abstract, which asserts a completed proof of the stated biconditional yet contains neither the proof steps, the precise definitions of the key notions (right uniserial, centrally essential, discrete valuation domain), nor any indication of where those definitions are taken from the literature. Consequently the soundness of the central claim cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The full manuscript (arXiv:1907.00946) contains the complete proofs, definitions, and references; the abstract is only a summary. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] The provided text consists solely of the abstract, which asserts a completed proof of the stated biconditional yet contains neither the proof steps, the precise definitions of the key notions (right uniserial, centrally essential, discrete valuation domain), nor any indication of where those definitions are taken from the literature. Consequently the soundness of the central claim cannot be assessed.

    Authors: The referee was evidently supplied only the abstract. The submitted manuscript includes the full proofs of the stated equivalence, drawing on standard definitions: a ring is right uniserial if its right ideals are linearly ordered by inclusion; centrally essential means the center is an essential submodule of the ring as a bimodule; discrete valuation domains are the usual commutative ones. These are referenced to the literature on uniserial rings and modules (e.g., standard texts on Artinian and Noetherian rings). The proofs proceed by first handling the commutative case, then showing that non-commutative examples must be Artinian on both sides, with explicit constructions of non-commutative Artinian uniserial centrally essential rings given in the body. If the journal requires an expanded introduction repeating these definitions, we can add one. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes a biconditional classification of right uniserial right Noetherian centrally essential rings via standard ring axioms and lemmas drawn from prior literature. No fitted parameters, empirical predictions, self-referential definitions, or load-bearing self-citations appear; the central equivalence is derived directly from the given definitions of uniserial, Noetherian, Artinian, discrete valuation domain, and centrally essential rings without reduction to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is a pure characterization theorem resting on the ordinary axioms of associative rings with identity and on the standard definitions of the listed ring-theoretic properties; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (2)
  • standard math Rings are associative with identity
    Invoked implicitly throughout the statement of the theorem.
  • domain assumption Uniserial, Noetherian, Artinian, discrete valuation domain, and centrally essential are defined exactly as in the prior literature
    The equivalence is stated using these terms without redefinition in the abstract.

pith-pipeline@v0.9.0 · 5610 in / 1322 out tokens · 36182 ms · 2026-05-25T11:03:51.908582+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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    Centrally essential rings (Russian), Diskretnaya Matematika

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    work page doi:10.1007/s13366-019-00447-w

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    Cayley-Dickson Process and Centrally Essential Rings

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    work page doi:10.1142/s0219498819502293

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    Semidistributive Modules and Rings, Springer Neth er- lands (Kluwer), Dordrecht-Boston-London, 1998, 352 pp

    Tuganbaev A. Semidistributive Modules and Rings, Springer Neth er- lands (Kluwer), Dordrecht-Boston-London, 1998, 352 pp. 7

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