Automorphism-Liftable Modules

Askar Tuganbaev

arxiv: 1907.00947 · v1 · pith:S7YPVAWEnew · submitted 2019-07-01 · 🧮 math.RA

Automorphism-Liftable Modules

Askar Tuganbaev This is my paper

Pith reviewed 2026-05-25 11:00 UTC · model grok-4.3

classification 🧮 math.RA

keywords automorphism-liftable modulestorsion moduleshereditary Noetherian prime ringsDedekind prime ringsmodule classificationnoncommutative ring theory

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The pith

All automorphism-liftable torsion modules over non-primitive hereditary Noetherian prime rings receive a complete description.

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

The paper establishes a full classification of automorphism-liftable torsion modules over non-primitive hereditary Noetherian prime rings. It separately examines the structure of automorphism-liftable non-torsion modules over Dedekind prime rings that need not be commutative. A reader cares because the work isolates precisely which modules allow automorphisms to lift while respecting the torsion or non-torsion condition. This supplies concrete structural information inside two standard classes of rings used in module theory.

Core claim

In this paper, we describe all automorphism-liftable torsion modules over non-primitive hereditary Noetherian prime rings. We also study automorphism-liftable non-torsion modules over not necessarily commutative Dedekind prime rings.

What carries the argument

automorphism-liftable modules (modules in which automorphisms of quotients lift to the module itself)

If this is right

Every such torsion module belongs to one of the explicitly listed families.
The classification distinguishes the torsion case from the non-torsion case over Dedekind prime rings.
The same lifting property can be checked directly against the listed forms rather than verified abstractly.

Where Pith is reading between the lines

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

The same lifting condition may admit similar descriptions when the ring is weakened to other hereditary or prime classes.
The listed modules may have endomorphism rings whose structure follows immediately from the classification.
One could test whether the listed forms remain automorphism-liftable when the module is placed over a larger ring containing the original one.

Load-bearing premise

The definitions of automorphism-liftable, torsion, and the listed ring classes remain exactly as fixed in the existing literature.

What would settle it

An explicit automorphism-liftable torsion module over a non-primitive hereditary Noetherian prime ring whose structure lies outside the forms given in the description.

read the original abstract

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

Tuganbaev classifies automorphism-liftable torsion modules over non-primitive hereditary Noetherian prime rings and studies the non-torsion case over Dedekind primes.

read the letter

Hey colleague, the punchline with this paper is that Tuganbaev classifies all automorphism-liftable torsion modules over non-primitive hereditary Noetherian prime rings and studies the non-torsion modules over Dedekind prime rings. This is new because it supplies an explicit description in those settings, extending earlier results on liftable modules without reducing to a prior theorem. The work uses fixed standard definitions for the module property and the ring classes, so the claim stays grounded. The paper does well by keeping the argument descriptive and free of circular derivations or extra parameters. It organizes the modules in a narrow but established part of noncommutative ring theory. Soft spots are minor. The abstract alone leaves the proofs out of view, so exhaustiveness and any derivation details cannot be checked directly. Nothing in the setup points to an internal inconsistency or unsupported step. This paper is for specialists already working on modules over prime rings, especially hereditary Noetherian or Dedekind ones. A reader in that subfield gets the concrete descriptions. It deserves peer review as a solid classification result that belongs in the literature even with a limited audience.

Referee Report

0 major / 3 minor

Summary. The manuscript claims to describe all automorphism-liftable torsion modules over non-primitive hereditary Noetherian prime rings and to study automorphism-liftable non-torsion modules over (not necessarily commutative) Dedekind prime rings.

Significance. If the classification is exhaustive and the proofs are correct, the result supplies a concrete description of modules satisfying a specific lifting property for automorphisms, which is a modest but useful addition to the literature on modules over hereditary Noetherian prime rings and Dedekind primes.

minor comments (3)

[Abstract] The abstract states the main results but does not indicate the structure of the classification (e.g., whether the torsion modules are direct sums of cyclics or have a specific form); adding one sentence summarizing the form of the modules would improve readability.
[Introduction] Standard definitions of 'automorphism-liftable' and the ring classes are assumed known; a brief recall or precise reference to the literature definitions in §1 would help readers who are not specialists in the area.
Ensure that any examples or counter-examples used to show that the listed modules are indeed automorphism-liftable are placed immediately after the statement of the classification theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

This is a classification theorem in ring and module theory. The paper states definitions of 'automorphism-liftable' and the indicated ring classes up front as standard notions from the literature, then describes which modules satisfy the property. No equations, fitted parameters, predictions, or derivations appear that reduce by construction to the inputs. The argument rests on fixed external definitions rather than any self-referential loop or self-citation chain that bears the central claim.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, ad-hoc axioms, or invented entities; all content is presumed to rest on standard definitions from prior ring theory literature.

pith-pipeline@v0.9.0 · 5537 in / 1027 out tokens · 21428 ms · 2026-05-25T11:00:58.527968+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

Lifting of automorphisms of fact or modules // Commun

Abyzov A.N., Quynh T.C. Lifting of automorphisms of fact or modules // Commun. Algebra. - 2018. V. 46, no. 11. – P. 5073-5082

work page 2018
[2]

// Additi ve unit struc- ture of endomorphism rings and invariance of modules

Guil Asensio P.A., Quynh T.C., Srivastava A.K. // Additi ve unit struc- ture of endomorphism rings and invariance of modules. – Bull . Math. Sci. – 2017. – Vol. 7. – P. 229-246

work page 2017
[3]

Skew projective Abelian groups // Indag

Janakiraman S. Skew projective Abelian groups // Indag. Math. – 1973. V.76, no. 3. – P.233–236

work page 1973
[4]

London Math

Lenagan T.H., Bounded hereditary Noetherian prime ring s // J. London Math. Soc. – 1973. – Vol. 6. – P. 241–246

work page 1973
[5]

C., Robson J

McConnell J. C., Robson J. C. Noncommutative Noetherian Rings. New York: Wiley-Interscience, 1987

work page 1987
[6]

Mishina A. P. On automorphisms and endomorphisms of Abel ian groups // Moscow University Mathematics Bulletin. – 1972. – no. 1. – P. 62–66

work page 1972
[7]

Selvaraj C., Santhakumar A. S. Automorphism liftable mo dules // Com- ment. Math. Univ. Carolin. – 2018. – V. 59, no. 1. – 35-44

work page 2018
[8]

Singh S., Quasi-injective and quasi-projective module s over hereditary Noetherian prime rings // Canad. J. Math. – 1974. – Vol. 26, no . 5. – P. 1173–1185. 11

work page 1974
[9]

Modules over hereditary Noetherian prime rings // Can

Singh S. Modules over hereditary Noetherian prime rings // Can. J. Math. 1975. V. 27, No. 4. P. 867–883

work page 1975
[10]

Tuganbaev A. A. The structure of modules close to projec tive modules // Sbornik: Mathematics – 1979. – V. 35, no. 2. – P. 219-228

work page 1979
[11]

Tuganbaev A. A. Quasi-projective modules // Sib. math. j. 1980. Vol. 21, no. 3. P. 446–450

work page 1980
[12]

Tuganbaev A. A. Semiprojective modules // Sib. math. j. 1980. Vol. 21, no. 5. P. 725–728

work page 1980
[13]

Tuganbaev A. A. Modules over bounded Dedekind prime rin gs // Sbornik: Mathematics – 2001. – V. 192, no. 5. – P. 705-724

work page 2001
[14]

Tuganbaev A. A. Automorphisms of submodules and their e xtensions // Discrete Math. Appl. – 2013. – Vol. 23, no. 1. – P. 115-124

work page 2013
[15]

Tuganbaev A. A. Automorphism-extendable and endomorp hism- extendable modules // J. Math. Sci. (New York) – To appear

work page
[16]

Foundations of Module and Ring Theory

Wisbauer R. Foundations of Module and Ring Theory. Phil adelphia: Gordon and Breach, 1991. 12

work page 1991

[1] [1]

Lifting of automorphisms of fact or modules // Commun

Abyzov A.N., Quynh T.C. Lifting of automorphisms of fact or modules // Commun. Algebra. - 2018. V. 46, no. 11. – P. 5073-5082

work page 2018

[2] [2]

// Additi ve unit struc- ture of endomorphism rings and invariance of modules

Guil Asensio P.A., Quynh T.C., Srivastava A.K. // Additi ve unit struc- ture of endomorphism rings and invariance of modules. – Bull . Math. Sci. – 2017. – Vol. 7. – P. 229-246

work page 2017

[3] [3]

Skew projective Abelian groups // Indag

Janakiraman S. Skew projective Abelian groups // Indag. Math. – 1973. V.76, no. 3. – P.233–236

work page 1973

[4] [4]

London Math

Lenagan T.H., Bounded hereditary Noetherian prime ring s // J. London Math. Soc. – 1973. – Vol. 6. – P. 241–246

work page 1973

[5] [5]

C., Robson J

McConnell J. C., Robson J. C. Noncommutative Noetherian Rings. New York: Wiley-Interscience, 1987

work page 1987

[6] [6]

Mishina A. P. On automorphisms and endomorphisms of Abel ian groups // Moscow University Mathematics Bulletin. – 1972. – no. 1. – P. 62–66

work page 1972

[7] [7]

Selvaraj C., Santhakumar A. S. Automorphism liftable mo dules // Com- ment. Math. Univ. Carolin. – 2018. – V. 59, no. 1. – 35-44

work page 2018

[8] [8]

Singh S., Quasi-injective and quasi-projective module s over hereditary Noetherian prime rings // Canad. J. Math. – 1974. – Vol. 26, no . 5. – P. 1173–1185. 11

work page 1974

[9] [9]

Modules over hereditary Noetherian prime rings // Can

Singh S. Modules over hereditary Noetherian prime rings // Can. J. Math. 1975. V. 27, No. 4. P. 867–883

work page 1975

[10] [10]

Tuganbaev A. A. The structure of modules close to projec tive modules // Sbornik: Mathematics – 1979. – V. 35, no. 2. – P. 219-228

work page 1979

[11] [11]

Tuganbaev A. A. Quasi-projective modules // Sib. math. j. 1980. Vol. 21, no. 3. P. 446–450

work page 1980

[12] [12]

Tuganbaev A. A. Semiprojective modules // Sib. math. j. 1980. Vol. 21, no. 5. P. 725–728

work page 1980

[13] [13]

Tuganbaev A. A. Modules over bounded Dedekind prime rin gs // Sbornik: Mathematics – 2001. – V. 192, no. 5. – P. 705-724

work page 2001

[14] [14]

Tuganbaev A. A. Automorphisms of submodules and their e xtensions // Discrete Math. Appl. – 2013. – Vol. 23, no. 1. – P. 115-124

work page 2013

[15] [15]

Tuganbaev A. A. Automorphism-extendable and endomorp hism- extendable modules // J. Math. Sci. (New York) – To appear

work page

[16] [16]

Foundations of Module and Ring Theory

Wisbauer R. Foundations of Module and Ring Theory. Phil adelphia: Gordon and Breach, 1991. 12

work page 1991