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arxiv: 2606.29060 · v2 · pith:UCLW4IEN · submitted 2026-06-27 · math.RA

The PI property of skew PBW extensions

James G\'omez , Claudia Gallego This is my paper

Reviewed by Pith2026-07-03 22:53 UTCgrok-4.3pith:UCLW4IENopen to challenge →

classification math.RA
keywords skew PBW extensionsPI propertynontrivial centerprime PI-algebrabijective extensionsnoncommutative polynomial ringspolynomial identities
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0 comments X

The pith

Every bijective skew PBW extension over a prime PI-algebra has a nontrivial center.

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

The paper shows that any bijective skew PBW extension built on a prime PI-algebra must have a center strictly larger than the scalars. This property makes it possible to decide whether the extension obeys a polynomial identity once the center is known explicitly. The same center fact is used to study PI behavior for certain algebras over fields of positive characteristic.

Core claim

Every bijective skew PBW extension over a prime PI-algebra has nontrivial center. This fact allows determining, from the known description of the center in several classes of examples, whether such extensions satisfy a polynomial identity. Building on results of Brown and Zhang, the PI property of certain K-algebras over fields of positive characteristic is investigated.

What carries the argument

The bijective skew PBW extension, a twisted polynomial ring whose multiplication rules are given by an automorphism and a derivation that together preserve bijectivity.

If this is right

  • In any class of bijective skew PBW extensions whose centers have already been computed, one can immediately conclude whether the ring is PI.
  • The result applies directly to the standard families of examples where center descriptions are known.
  • Certain K-algebras over positive-characteristic fields inherit a PI property from the center analysis.
  • Nontrivial center supplies a concrete obstruction to the extension being a domain or simple ring in the usual ways.

Where Pith is reading between the lines

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

  • The same center argument may apply to other families of twisted polynomial rings that satisfy analogous bijectivity and primeness hypotheses.
  • Explicit computation of centers in new families would give immediate PI verdicts without separate identity checks.
  • If a counter-example exists it must fail either bijectivity or the primeness/PI assumption on the base ring.

Load-bearing premise

The skew PBW extension must be bijective and the base algebra must be a prime PI-algebra.

What would settle it

An explicit bijective skew PBW extension over a prime PI-algebra whose center consists only of scalars would disprove the claim.

read the original abstract

In this article we study the polynomial identity (PI) property of skew PBW extensions. We show that every bijective skew PBW extension over a prime PI-algebra has nontrivial center. This fact allows us to determine, from the known description of the center in several classes of examples, whether such extensions satisfy a polynomial identity. Furthermore, building on results of Brown and Zhang \cite{BrownZhang2022}, we investigate the PI property of certain $\K$-algebras over fields of positive characteristic.

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

0 major / 3 minor

Summary. The manuscript proves that every bijective skew PBW extension of a prime PI-algebra has nontrivial center. This is applied to decide the PI property of several classes of such extensions by inspecting their centers from existing descriptions. The paper further studies the PI property of certain K-algebras over fields of positive characteristic, building on results of Brown and Zhang.

Significance. If the central result holds, it supplies a practical criterion linking the center to the PI property for bijective skew PBW extensions, a class of noncommutative rings of independent interest. The explicit use of known center descriptions in concrete families is a strength that makes the theorem immediately applicable. The positive-characteristic investigation extends prior work without overclaiming generality.

minor comments (3)
  1. The definition of 'nontrivial center' (whether it means strictly larger than the base field or something else) should be stated explicitly in the introduction or §2 when the main theorem is formulated.
  2. The statement that the result 'allows us to determine' the PI property from known centers would be strengthened by a brief table or list in §4 or §5 indicating which of the cited example classes satisfy the hypotheses.
  3. A short remark clarifying how bijectivity is used in the proof (e.g., to guarantee that certain automorphisms or derivations are invertible) would improve readability for readers unfamiliar with the skew PBW literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of the central result's applicability and the extension of prior work on positive-characteristic algebras. The recommendation for minor revision is noted. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central claim is that every bijective skew PBW extension over a prime PI-algebra has nontrivial center. This is stated as a result to be proved under explicit hypotheses (bijectivity of the extension plus the base ring being prime and PI). The abstract indicates the proof is self-contained for the main theorem and only invokes external Brown-Zhang results for a separate investigation in positive characteristic. No equations, definitions, or steps are shown that reduce the conclusion to a fitted parameter, a self-citation chain, or an ansatz imported from the authors' prior work. The conditions required for the conclusion are openly listed rather than derived from the result itself. This is the normal case of a non-circular derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities; full paper required to populate the ledger.

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Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    A. D. Bell. Notes on localization in noncommutative noetherian rings. Cuadernos de Álgebra 9, Universidad de Granada, 1989. 77 pp.; revised PDF version dated October 1997

    work page 1989

  2. [2]

    Bera and S

    S. Bera and S. Mukherjee. PI degree of quantized Weyl algebras.Proc. Indian Acad. Sci. Math. Sci., 133:18, 2023

    work page 2023

  3. [3]

    Bera and S

    S. Bera and S. Mukherjee. On simple modules over the quantum matrix algebra at roots of unity, 2025.https://arxiv.org/abs/2503.10394

    work page arXiv 2025

  4. [4]

    Bera and S

    S. Bera and S. Mukherjee.U + q (B2)and its representations, 2025.https://arxiv.org/abs/ 2503.21170

    work page arXiv 2025

  5. [5]

    R. Berger. The quantum Poincaré–Birkhoff–Witt theorem.Comm. Math. Phys., 143(2):215–234, 1992

    work page 1992

  6. [6]

    K. A. Brown and J. J. Zhang. Iterated Hopf Ore extensions in positive characteristic.J. Noncommut. Geom., 16(3):787–837, 2022

    work page 2022

  7. [7]

    P . A. A. B. Carvalho, S. A. Lopes, and J. Matczuk. Double Ore extensions versus iterated Ore extensions.Comm. Algebra, 39(8):2838–2848, 2011

    work page 2011

  8. [8]

    Cauchon.Les T-anneaux et les anneaux à identité polynomiale noethériens

    G. Cauchon.Les T-anneaux et les anneaux à identité polynomiale noethériens. Thèse, Université de Paris-Sud, Orsay, 1977

    work page 1977

  9. [9]

    K. Chan, J. Gaddis, R. Won, and J. J. Zhang. Ozone groups of Artin–Schelter regular algebras satisfying a polynomial identity.Math. Z., 310:69, 2025

    work page 2025

  10. [10]

    Daniano and J

    R. Daniano and J. Shapiro. Twisted polynomial rings satisfying a polynomial identity.J. Algebra, 92:116–127, 1985. 27

    work page 1985

  11. [11]

    De Concini and V

    C. De Concini and V . G. Kac. Representations of quantum groups at root of1. In A. Connes, M. Duflo, and R. Rentschler, editors,Operator Algebras, Unitary Representations, Enveloping Algebras and Invariant Theory, pages 472–506. Birkhäuser, Boston, 1990

    work page 1990

  12. [12]

    De Concini and C

    C. De Concini and C. Procesi. Quantum groups. In Giuseppe Zampieri and Andrea D’Agnolo, ed- itors,D-modules, Representation Theory, and Quantum Groups, volume 1565 ofLecture Notes in Mathe- matics, pages 31–140. Springer, Berlin, 1993

    work page 1993

  13. [13]

    M. Dehn. Über die Grundlagen der projektiven Geometrie und allgemeine Zahlsysteme.Math. Ann., 85(1):184–194, 1922

    work page 1922

  14. [14]

    Dixmier.Enveloping Algebras, volume 11 ofGraduate Studies in Mathematics

    J. Dixmier.Enveloping Algebras, volume 11 ofGraduate Studies in Mathematics. American Mathemat- ical Society, Providence, RI, 1996

    work page 1996

  15. [15]

    Fajardo, C

    W. Fajardo, C. Gallego, O. Lezama, A. Reyes, H. Suárez, and H. Venegas.Skew PBW Extensions: Ring and Module-theoretic Properties, Matrix and Gröbner Methods, and Applications. Algebra and Ap- plications. Springer International Publishing, 2021

    work page 2021

  16. [16]

    J. Gaddis. Two-parameter analogs of the Heisenberg enveloping algebra.Comm. Algebra, 44(11):4637–4653, 2016

    work page 2016

  17. [17]

    Gaddis and D

    J. Gaddis and D. Yee. A family of trivial ozone algebras, 2025.https://arxiv.org/abs/2512. 09766

    work page 2025

  18. [18]

    Gallego and O

    C. Gallego and O. Lezama. Gröbner bases for ideals ofσ–PBW extensions.Comm. Algebra, 39(1):50– 75, 2011

    work page 2011

  19. [19]

    K. R. Goodearl and Jr. Warfield, R. B.An Introduction to Noncommutative Noetherian Rings, volume 61 ofLondon Mathematical Society Student Texts. Cambridge University Press, 2 edition, 2004

    work page 2004

  20. [20]

    Gómez and C

    J. Gómez and C. Gallego. The PI property in algebras of polynomial type, 2026.https://arxiv. org/abs/2603.23724

    work page arXiv 2026

  21. [21]

    H. Haynal. PI degree parity inq-skew polynomial rings.J. Algebra, 319:4199–4221, 2008

    work page 2008

  22. [22]

    Jacobson.Collected mathematical papers

    N. Jacobson.Collected mathematical papers. Vol. 2. Contemporary Mathematicians. Birkhäuser Boston, Inc., Boston, MA, 1989. (1947–1965)

    work page 1989

  23. [23]

    Kaplansky

    I. Kaplansky. Rings with a polynomial identity.Bull. Amer. Math. Soc., 54:575–580, 1948

    work page 1948

  24. [24]

    Leroy and J

    A. Leroy and J. Matczuk. Ore extensions satisfying a polynomial identity.J. Algebra Appl., 2(3):257– 273, 2007

    work page 2007

  25. [25]

    Levandovskyy.Non-commutative Computer Algebra for Polynomial Algebras: Gröbner Bases, Appli- cations and Implementation

    V . Levandovskyy.Non-commutative Computer Algebra for Polynomial Algebras: Gröbner Bases, Appli- cations and Implementation. Doctoral thesis, Universität Kaiserslautern, Germany, 2005

    work page 2005

  26. [26]

    Lezama, J

    O. Lezama, J. Acosta, and A. Reyes. Prime ideals of skew PBW extensions.Rev. Un. Mat. Argentina, 56:39–55, 2015

    work page 2015

  27. [27]

    Lezama and A

    O. Lezama and A. Reyes. Some homological properties of skew PBW extensions.Comm. Algebra, 42:1200–1230, 2014

    work page 2014

  28. [28]

    Lezama and H

    O. Lezama and H. Venegas. Center of skew PBW extensions.Internat. J. Algebra Comput., 30(8):1625– 1650, 2020

    work page 2020

  29. [29]

    S. A. Lopes, H. Suárez, and Y. Suárez. Homological properties of some quantum Heisenberg alge- bras, 2025.https://arxiv.org/abs/2503.03978

    work page arXiv 2025

  30. [30]

    J. C. McConnell, J. C. Robson, and L. W. Small.Noncommutative Noetherian Rings, volume 30 of Graduate Studies in Mathematics. American Mathematical Society, 2001. 28

    work page 2001

  31. [31]

    S.-Q. Oh. Primitive ideals of the coordinate ring of quantum symplectic space.J. Algebra, 174(2):531– 552, 1995

    work page 1995

  32. [32]

    Pascaud and J

    J. Pascaud and J. Valette. Polynômes tordus à identité polynomiale.Comm. Algebra, 16:2415–2425, 1988

    work page 1988

  33. [33]

    E. N. Shirikov. Two-generated graded algebras.Algebra Discrete Math., 4(3):60–84, 2005

    work page 2005

  34. [34]

    H. Suárez. Koszulity for graded skew PBW extensions.Comm. Algebra, 45:4569–4580, 2017

    work page 2017

  35. [35]

    W. Wagner. Über die Grundlagen der projektiven Geometrie und allgemeine Zahlensysteme.Math. Ann., 113(1):528–567, 1937

    work page 1937

  36. [36]

    S. L. Woronowicz. Twisted SU(2) group. An example of a noncommutative differential calculus. Publ. Res. Inst. Math. Sci., 23:117–181, 1987

    work page 1987

  37. [37]

    H. Zhang. The representations of the coordinate ring of the quantum symplectic space.J. Pure Appl. Algebra, 150(1):95–106, 2000

    work page 2000

  38. [38]

    J. J. Zhang and J. Zhang. Double Ore extensions.J. Pure Appl. Algebra, 212(12):2668–2690, 2008. J.G.: Escuela de Matemáticas y Estadística, Universidad Pedagógica y Tecnológica de Colombia, Tunja, Colombia james.gomez@uptc.edu.co C.G.: Departamento de Matemáticas, Pontificia Universidad Javeriana, Bogotá, Colombia gallegoj.cm@javeriana.edu.co 29

    work page 2008