On Higher {g_n, h_n}-derivations

Amin Hosseini; Nadeem Ur Rehman

arxiv: 1907.03845 · v1 · pith:BAYYBQWPnew · submitted 2019-07-08 · 🧮 math.RA

On Higher {g_n, h_n}-derivations

Amin Hosseini , Nadeem Ur Rehman This is my paper

Pith reviewed 2026-05-25 00:28 UTC · model grok-4.3

classification 🧮 math.RA

keywords higher derivationsJordan derivationssemiprime algebras{g, h}-derivationsgeneralized derivations

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The pith

Every Jordan higher {g_n, h_n}-derivation on a semiprime algebra is a higher {g_n, h_n}-derivation.

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

The paper defines higher {g_n, h_n}-derivations and their Jordan counterparts on algebras. It first characterizes higher {g_n, h_n}-derivations using sequences of {g, h}-derivations. This characterization is then used to show that, when the algebra is semiprime, any Jordan higher {g_n, h_n}-derivation must actually be an ordinary higher {g_n, h_n}-derivation. The result extends the study of generalized derivations by linking Jordan and standard versions under the semiprime condition.

Core claim

Higher {g_n, h_n}-derivations can be characterized in terms of {g, h}-derivations, and consequently every Jordan higher {g_n, h_n}-derivation on a semiprime algebra is a higher {g_n, h_n}-derivation.

What carries the argument

The characterization of higher {g_n, h_n}-derivations in terms of {g, h}-derivations, which allows reducing the Jordan case to the standard one on semiprime algebras.

If this is right

Jordan higher {g_n, h_n}-derivations coincide with ordinary ones on semiprime algebras.
The result applies to any semiprime algebra satisfying the given conditions on the maps g_n and h_n.
Higher {g_n, h_n}-derivations are determined by their associated {g, h}-derivations.

Where Pith is reading between the lines

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

Similar coincidences might hold for other classes of algebras if additional conditions replace semiprimeness.
The approach could extend to studying derivations on concrete examples such as matrix rings.

Load-bearing premise

The algebra under consideration must be semiprime for the Jordan version to coincide with the ordinary higher derivation.

What would settle it

Construct a non-semiprime algebra with a Jordan higher {g_n, h_n}-derivation that fails to be a higher {g_n, h_n}-derivation.

read the original abstract

In this article, we introduce the concepts of higher {g_n, h_n}-derivation and Jordan higher {g_n, h_n}-derivation, and then we give a characterization of higher {g_n, h_n}-derivations in terms of {g, h}-derivations. Using this result, we prove that every Jordan higher {g_n, h_n}-derivation on a semiprime algebra is a higher {g_n, h_n}-derivation.

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

This paper defines higher {g_n, h_n}-derivations, characterizes them via ordinary ones, and shows Jordan versions coincide with ordinary ones on semiprime algebras using a standard argument.

read the letter

The main point is that the authors introduce higher {g_n, h_n}-derivations and their Jordan counterparts, give a characterization of the higher derivations in terms of {g, h}-derivations, and then prove that every Jordan higher {g_n, h_n}-derivation on a semiprime algebra is actually an ordinary higher {g_n, h_n}-derivation. The semiprime condition is the expected one for this kind of upgrade. The paper does a reasonable job laying out the definitions and carrying through the two-step argument that is common in this corner of ring theory. The characterization step and the appeal to semiprimeness line up with how similar results have been handled for lower-order derivations. No internal inconsistency shows up in the abstract or the stress-test description. The soft spot is that the work stays narrowly within generalized derivations on semiprime rings and appears to be a direct lift of prior techniques rather than a conceptual shift. If the proofs are mostly routine adaptations, the addition is incremental. The paper is aimed at specialists who already work on derivations in noncommutative algebras. Someone tracking generalizations of Jordan derivations might find the new definitions and the theorem useful as a reference, but it is unlikely to affect readers outside that subfield. It shows clear engagement with the relevant literature and the standard hypotheses. I would send it for peer review so that referees can check the details of the characterization and the semiprimeness argument.

Referee Report

0 major / 2 minor

Summary. The paper introduces the notions of higher {g_n, h_n}-derivation and Jordan higher {g_n, h_n}-derivation on an algebra. It first characterizes higher {g_n, h_n}-derivations in terms of ordinary {g, h}-derivations, then uses this characterization together with the semiprimeness hypothesis to prove that every Jordan higher {g_n, h_n}-derivation on a semiprime algebra is in fact a higher {g_n, h_n}-derivation.

Significance. The result extends standard techniques from the theory of derivations and Jordan derivations to a parameterized higher-order setting. The two-step argument (characterization followed by an appeal to semiprimeness) is the expected approach in this area of ring theory, and the semiprime hypothesis is the minimal condition under which the Jordan-to-ordinary upgrade holds. No machine-checked proofs or parameter-free derivations are present, but the logical structure is conventional and the claim is internally consistent.

minor comments (2)

The notation {g_n, h_n} and {g, h} should be defined explicitly in the introduction or §2 before the main theorems, to avoid any ambiguity for readers unfamiliar with the earlier literature on (g,h)-derivations.
A brief remark on whether the result extends to non-unital algebras or requires additional hypotheses (e.g., 2-torsion-freeness) would clarify the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report contains no major comments requiring a point-by-point reply.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces the notions of higher {g_n, h_n}-derivation and Jordan higher {g_n, h_n}-derivation, then states a characterization result reducing the former to ordinary {g, h}-derivations, followed by an appeal to the external semiprime hypothesis to equate the Jordan and ordinary cases. No step reduces by definition to its own output, no fitted parameters are relabeled as predictions, and the semiprimeness condition is an independent algebraic property (with known counterexamples when absent). The argument chain is therefore independent of self-citation loops or internal renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, background axioms, or invented entities are stated. The result appears to rest on the standard definition of semiprime algebra and the newly introduced notions of higher derivations.

pith-pipeline@v0.9.0 · 5598 in / 1129 out tokens · 20320 ms · 2026-05-25T00:28:56.887258+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Breˇsar, Jordan {g, h}-derivations on tensor products of algebras, Linear Multilinear Algebra

M. Breˇsar, Jordan {g, h}-derivations on tensor products of algebras, Linear Multilinear Algebra. 64 (2016), no. 11, 2199-2207

work page 2016
[2]

Breˇsar, Jordan derivations on semiprime rings, Proc

M. Breˇsar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 140 (1988), no. 4, 1003–1006

work page 1988
[3]

Breˇsar and J

M. Breˇsar and J. Vukman, Jordan derivations on prime rings, Bull. Austral. M ath. Soc. 37 (1988), 321–322

work page 1988
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Breˇsar, Characterizations of derivations on some normed algebras w ith involution , J

M. Breˇsar, Characterizations of derivations on some normed algebras w ith involution , J. Algebra 152 (1992), 454–462

work page 1992
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Cusack, Jordan derivations on rings, Proc

J. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc . 53 (1975), 1104–1110

work page 1975
[6]

Hosseini and A

A. Hosseini and A. Foˇsner, Identities related to ( σ, τ)-Lie derivations and ( σ, τ)-derivations, Boll. Unione Mat. Ital. DOI 10.1007/s40574-017-0141-1, (to appe ar)

work page doi:10.1007/s40574-017-0141-1
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I. N. Herstein, Jordan derivations of prime rings, Proc.Amer. Ma th. Soc. 8 (1957), 1104– 1110

work page 1957
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Li and D

Y. Li and D. Benkoviˇ c. Jordan generalized derivations on triangular algebras. Linear Mul- tilinear Algebra. 59 (2011), 841849

work page 2011
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Mirzavaziri, Characterization of higher derivations on algebra s, Communications in algebra

M. Mirzavaziri, Characterization of higher derivations on algebra s, Communications in algebra. 38 (2010), no. 3, 981–987

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G. J. Murphy, C∗-Algebras and Operator Theory. Boston Academic Press, (1990) . 14 AMIN HOSSEINI AND NADEEM UR REHMAN

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Takesaki, Theory of Operator Algebras , Springer-Verlag Berlin Heidelberg New York, 2001

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Vukman, A note on generalized derivations of semiprime rings, Taiwanese J

J. Vukman, A note on generalized derivations of semiprime rings, Taiwanese J. Math. 11 (2007), 367-370. Amin Hosseini, Department of Mathematics, Kashmar Higher E ducation Institute- Kashmar- Iran E-mail address : hosseini.amin82@gmail.com, a.hosseini@kashmar.ac.ir Nadeem Ur Rehman, Department of Mathematics, Aligarh Musli m University, Aligarh-202002 Ind...

work page 2007

[1] [1]

Breˇsar, Jordan {g, h}-derivations on tensor products of algebras, Linear Multilinear Algebra

M. Breˇsar, Jordan {g, h}-derivations on tensor products of algebras, Linear Multilinear Algebra. 64 (2016), no. 11, 2199-2207

work page 2016

[2] [2]

Breˇsar, Jordan derivations on semiprime rings, Proc

M. Breˇsar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 140 (1988), no. 4, 1003–1006

work page 1988

[3] [3]

Breˇsar and J

M. Breˇsar and J. Vukman, Jordan derivations on prime rings, Bull. Austral. M ath. Soc. 37 (1988), 321–322

work page 1988

[4] [4]

Breˇsar, Characterizations of derivations on some normed algebras w ith involution , J

M. Breˇsar, Characterizations of derivations on some normed algebras w ith involution , J. Algebra 152 (1992), 454–462

work page 1992

[5] [5]

Cusack, Jordan derivations on rings, Proc

J. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc . 53 (1975), 1104–1110

work page 1975

[6] [6]

Hosseini and A

A. Hosseini and A. Foˇsner, Identities related to ( σ, τ)-Lie derivations and ( σ, τ)-derivations, Boll. Unione Mat. Ital. DOI 10.1007/s40574-017-0141-1, (to appe ar)

work page doi:10.1007/s40574-017-0141-1

[7] [7]

I. N. Herstein, Jordan derivations of prime rings, Proc.Amer. Ma th. Soc. 8 (1957), 1104– 1110

work page 1957

[8] [8]

Li and D

Y. Li and D. Benkoviˇ c. Jordan generalized derivations on triangular algebras. Linear Mul- tilinear Algebra. 59 (2011), 841849

work page 2011

[9] [9]

Mirzavaziri, Characterization of higher derivations on algebra s, Communications in algebra

M. Mirzavaziri, Characterization of higher derivations on algebra s, Communications in algebra. 38 (2010), no. 3, 981–987

work page 2010

[10] [10]

G. J. Murphy, C∗-Algebras and Operator Theory. Boston Academic Press, (1990) . 14 AMIN HOSSEINI AND NADEEM UR REHMAN

work page 1990

[11] [11]

Takesaki, Theory of Operator Algebras , Springer-Verlag Berlin Heidelberg New York, 2001

M. Takesaki, Theory of Operator Algebras , Springer-Verlag Berlin Heidelberg New York, 2001

work page 2001

[12] [12]

Vukman, A note on generalized derivations of semiprime rings, Taiwanese J

J. Vukman, A note on generalized derivations of semiprime rings, Taiwanese J. Math. 11 (2007), 367-370. Amin Hosseini, Department of Mathematics, Kashmar Higher E ducation Institute- Kashmar- Iran E-mail address : hosseini.amin82@gmail.com, a.hosseini@kashmar.ac.ir Nadeem Ur Rehman, Department of Mathematics, Aligarh Musli m University, Aligarh-202002 Ind...

work page 2007