pith. sign in

arxiv: 2605.20008 · v2 · pith:HKHI264Bnew · submitted 2026-05-19 · 🧮 math.RA · math.GR

Central idempotents in group-graded rings

Johan \"Oinert This is my paper

Pith reviewed 2026-06-30 17:30 UTC · model grok-4.3

classification 🧮 math.RA math.GR
keywords group-graded ringscentral idempotentsfinite supporttorsion-free groupsnon-unital ringssemigroup-graded ringsLeavitt path rings
0
0 comments X

The pith

Every nonzero central idempotent in a G-graded ring has finite support when G is abelian or the grading satisfies a one-sided non-annihilation condition.

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

The paper proves that in a ring graded by a group G, any nonzero central idempotent must involve only finitely many group degrees under two sets of assumptions. When G itself is abelian this holds unconditionally. When G is arbitrary the grading must obey a one-sided non-annihilation rule that prevents nonzero homogeneous pieces from annihilating one another from one side. Under either hypothesis, if G is moreover torsion-free then the idempotent is forced to lie entirely inside the degree-zero component. The argument applies directly to possibly non-unital and non-commutative graded rings and recovers earlier statements about group rings and crossed products as special cases.

Core claim

We show that every nonzero central idempotent in R has finite support group in two broad settings: when G is abelian, and when G is arbitrary but the grading satisfies a certain one-sided non-annihilation condition on nonzero homogeneous elements. In particular, under the respective hypotheses, if G is torsion-free, then every central idempotent lies in the principal component of the grading.

What carries the argument

The support group of a central idempotent, consisting of those group elements whose corresponding homogeneous components are nonzero.

Load-bearing premise

When G is non-abelian the grading must satisfy the one-sided non-annihilation condition on nonzero homogeneous elements.

What would settle it

Exhibit a torsion-free abelian group G, a G-graded ring R, and a nonzero central idempotent whose support is infinite.

read the original abstract

Let $G$ be a group and let $R$ be a $G$-graded ring. We show that every nonzero central idempotent in $R$ has finite support group in two broad settings: when $G$ is abelian, and when $G$ is arbitrary but the grading satisfies a certain one-sided non-annihilation condition on nonzero homogeneous elements. In particular, under the respective hypotheses, if $G$ is torsion-free, then every central idempotent lies in the principal component of the grading. Our results generalize earlier results by H. Bass, R. G. Burns, and A. A. Bovdi--S. V. Mihovski, from group rings and crossed products, to non-commutative, possibly non-unital, group-graded rings. We demonstrate the utility of our results by applying them to semigroup-graded rings, Leavitt path rings, fractional skew monoid rings, partial skew group rings, and algebraic Cuntz-Pimsner 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.

Referee Report

2 major / 1 minor

Summary. The paper shows that every nonzero central idempotent in a G-graded ring R has finite support group when G is abelian or when the grading satisfies a one-sided non-annihilation condition on nonzero homogeneous elements. Under these hypotheses, if G is torsion-free then every such idempotent lies in the principal component. The results generalize theorems of Bass, Burns, and Bovdi-Mihovski from group rings and crossed products to non-commutative, possibly non-unital graded rings, and are applied to semigroup-graded rings, Leavitt path rings, fractional skew monoid rings, partial skew group rings, and algebraic Cuntz-Pimsner rings.

Significance. If the derivations hold, the work supplies a useful extension of classical results on idempotents to a wider class of graded rings, including non-unital and non-commutative examples, with concrete applications that illustrate utility in contemporary ring-theoretic settings.

major comments (2)
  1. [Abstract] Abstract: the one-sided non-annihilation condition is introduced as sufficient for the finite-support conclusion when G is non-abelian, but no counter-example is supplied showing that the conclusion can fail when the condition is dropped, nor is necessity of the hypothesis established; this leaves the load-bearing status of the condition for the non-abelian case unclear.
  2. [Abstract] Abstract: the central claims are asserted to rest on proofs that are not sketched even at the level of lemmas or key steps, so the passage from the non-annihilation hypothesis to finite support cannot be inspected for circularity or hidden assumptions.
minor comments (1)
  1. The abstract lists five classes of rings to which the results are applied but gives no indication of the specific corollaries obtained in each case; a short sentence per application would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We respond to each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the one-sided non-annihilation condition is introduced as sufficient for the finite-support conclusion when G is non-abelian, but no counter-example is supplied showing that the conclusion can fail when the condition is dropped, nor is necessity of the hypothesis established; this leaves the load-bearing status of the condition for the non-abelian case unclear.

    Authors: The manuscript establishes the one-sided non-annihilation condition as a sufficient hypothesis for the finite-support conclusion in the non-abelian case; it does not assert or prove necessity. To clarify the role of the hypothesis, we will revise the abstract and add a sentence in the introduction explicitly noting that the condition is sufficient (generalizing the abelian case, where it is not required) but that we make no claim of necessity. We do not currently have an explicit counterexample demonstrating failure without the condition. revision: partial

  2. Referee: [Abstract] Abstract: the central claims are asserted to rest on proofs that are not sketched even at the level of lemmas or key steps, so the passage from the non-annihilation hypothesis to finite support cannot be inspected for circularity or hidden assumptions.

    Authors: Abstracts conventionally omit proof sketches. The full manuscript contains the detailed arguments: preliminary results on supports and homogeneous elements appear in Section 2, and the main theorems (3.1 for the abelian case and 4.2 for the non-abelian case with the non-annihilation condition) proceed by assuming an idempotent with infinite support and deriving a contradiction via the annihilation property on homogeneous components. To improve readability, we will insert a short proof-outline paragraph in the introduction. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is a standard generalization under explicit hypotheses

full rationale

The paper states two theorems under explicitly listed hypotheses (G abelian, or G arbitrary with one-sided non-annihilation on homogeneous elements) and proves finite support for central idempotents, generalizing independent prior results by Bass, Burns, and Bovdi-Mihovski. No self-citations appear as load-bearing steps, no parameters are fitted then renamed as predictions, and no equation reduces to a prior definition by construction. The non-annihilation condition is introduced as an assumption rather than derived from the conclusion, so the derivation chain remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard axioms of ring theory and group theory together with the two stated hypotheses on G or the grading; no free parameters or new postulated entities appear.

axioms (1)
  • standard math Standard axioms of associative rings, groups, and G-gradings
    The paper invokes the usual definitions of graded rings, central idempotents, and support of elements.

pith-pipeline@v0.9.1-grok · 5694 in / 1202 out tokens · 52930 ms · 2026-06-30T17:30:59.067150+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Graded differential polynomial rings

    math.RA 2026-06 unverdicted novelty 6.0

    Characterizes when differential polynomial rings R[t;δ] admit compatible Γ-gradings and proves graded analogues of simplicity, primeness, and Noetherianity theorems.

Reference graph

Works this paper leans on

20 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    Abrams, P

    G. Abrams, P. Ara, and M. Siles Molina,Leavitt path algebras, Lecture Notes in Math. 2191, Springer, London (2017)

    2017

  2. [2]

    P. Ara, M. A. González-Barroso, K. R. Goodearl, and E. Pardo, Fractional skew monoid rings,J. Algebra278(2004), no. 1, 104–126

    2004

  3. [3]

    Bass, Euler characteristics and characters of discrete groups,Invent

    H. Bass, Euler characteristics and characters of discrete groups,Invent. Math.35(1976), 155–196

    1976

  4. [4]

    A. A. Bovdi and S. V. Mihovski, Idempotents of crossed products,Dokl. Akad. Nauk SSSR195 (1970), 263–265

    1970

  5. [5]

    A. A. Bovdi and S. V. Mihovski, Algebraic elements of crossed products, Rings, modules and radicals (Proc. Colloq., Keszthely, 1971), pp. 103–116,Colloq. Math. Soc. János Bolyai6, North-Holland Publishing Co., Amsterdam-London (1973)

    1971

  6. [6]

    R. G. Burns, Central idempotents in group rings,Canad. Math. Bull.13(1970), 527–528

    1970

  7. [7]

    T. M. Carlsen and E. Ortega, Algebraic Cuntz-Pimsner rings,Proc. Lond. Math. Soc. (3)103(2011), no. 4, 601–653

    2011

  8. [8]

    A. H. Clifford and G. B. Preston,The algebraic theory of semigroups. Vol. I.Math. Surveys, No. 7, American Mathematical Society, Providence, RI (1961). CENTRAL IDEMPOTENTS IN GROUP-GRADED RINGS 11

    1961

  9. [9]

    de Cornulier, Infinite conjugacy classes in groups acting on trees,Groups Geom

    Y. de Cornulier, Infinite conjugacy classes in groups acting on trees,Groups Geom. Dyn.3(2009), no. 2, 267–277

    2009

  10. [10]

    J. L. Dorroh, Concerning adjunctions to algebras,Bull. Amer. Math. Soc.38(1932), no. 2, 85–88

    1932

  11. [11]

    Kirby, Idempotents in a graded ring,J

    D. Kirby, Idempotents in a graded ring,J. London Math. Soc. (2)8(1974), 375–376

    1974

  12. [12]

    Lännström, P

    D. Lännström, P. Lundström, J. Öinert, and S. Wagner, Prime group graded rings with applications to partial crossed products and Leavitt path algebras,J. Pure Appl. Algebra229(2025), no. 1, Paper No. 107842, 34 pp

    2025

  13. [13]

    Lännström, P

    D. Lännström, P. Lundström, J. Öinert, and S. Wagner, Nonunital prime rings graded by ordered groups, to appear in Bull. Aust. Math. Soc., preprint arXiv:2510.26734 [math.RA]

    work page arXiv

  14. [14]

    Lee and E

    P.-H. Lee and E. R. Puczyłowski, On prime ideals and radicals of polynomial rings and graded rings, J. Pure Appl. Algebra218(2014), no. 2, 323–332

    2014

  15. [15]

    2-3, 104–110

    S.V.Mihovski, Centralalgebraicelementsinstrictlygradedrings,Serdica17(1991), no. 2-3, 104–110

    1991

  16. [16]

    Năstăsescu and F

    C. Năstăsescu and F. Van Oystaeyen,Methods of graded rings, Lecture Notes in Math. 1836, Springer- Verlag, Berlin (2004)

    2004

  17. [17]

    Nauwelaerts and F

    E. Nauwelaerts and F. Van Oystaeyen, Introducing crystalline graded algebras,Algebr. Represent. Theory11(2008), no. 2, 133–148

    2008

  18. [18]

    Group Theory 27(2024), no

    J.Öinert, Units, zero-divisorsand idempotents inrings gradedby torsion-freegroups,J. Group Theory 27(2024), no. 4, 789–811

    2024

  19. [19]

    Rudin and H

    W. Rudin and H. Schneider, Idempotents in group rings,Duke Math. J.31(1964), 585–602

    1964

  20. [20]

    Tominaga, On s-unital rings,Math

    H. Tominaga, On s-unital rings,Math. J. Okayama Univ.18(1976), no. 2, 117–134. Department of Mathematics and Natural Sciences, Blekinge Institute of Technology, SE-37179 Karlskrona, Sweden and Department of Engineering, University of Skövde, SE- 54128 Skövde, Sweden Email address:johan.oinert@bth.se

    1976