arxiv: 2604.17544 · v1 · submitted 2026-04-19 · 🧮 math.RA · math.FA

Recognition: unknown

A note on n-Jordan homomorphisms

M. El Azhari

Pith reviewed 2026-05-10 05:07 UTC · model grok-4.3

classification 🧮 math.RA math.FA

keywords n-Jordan homomorphismsJordan homomorphismsring homomorphismsanti-homomorphismsHerstein theoremring theory

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The pith

If every Jordan homomorphism from a unital ring A to B is a homomorphism or anti-homomorphism, then every n-Jordan homomorphism is an n-homomorphism or anti-n-homomorphism when char(B) exceeds n.

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

The paper shows that a preservation property known for ordinary Jordan homomorphisms between rings carries over to n-Jordan homomorphisms under stated conditions on the rings. When A has a unit and the characteristic of B is greater than n, the assumption that all Jordan homomorphisms are homomorphisms or anti-homomorphisms forces the same conclusion for n-Jordan homomorphisms. The argument invokes a variation of Herstein's theorem on n-Jordan homomorphisms to obtain this extension directly. A reader cares because the result reduces verification of higher-order multiplicative maps to the already-studied Jordan case in ring theory.

Core claim

By applying a variation of Herstein's theorem on n-Jordan homomorphisms, the paper deduces that for rings A with a unit and B with char(B) > n, if every Jordan homomorphism from A into B is a homomorphism or anti-homomorphism, then every n-Jordan homomorphism from A into B is an n-homomorphism or anti-n-homomorphism.

What carries the argument

The variation of Herstein's theorem on n-Jordan homomorphisms, which deduces the n-case from the ordinary Jordan case under the given ring conditions.

If this is right

The homomorphism or anti-homomorphism property for Jordan maps extends to the corresponding n-Jordan maps.
The extension holds for every pair of rings satisfying the unit and characteristic conditions.
G. An's result follows immediately as a corollary of the variation of Herstein's theorem.

Where Pith is reading between the lines

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

The same variation of Herstein's theorem could connect other multiplicative preservation properties across different powers in rings of high characteristic.
Explicit checks on matrix rings or polynomial rings might reveal when n-Jordan maps coincide exactly with ordinary homomorphisms.
The reduction suggests that many ring-mapping properties scale with the exponent n once the base Jordan case is controlled.

Load-bearing premise

The variation of Herstein's theorem on n-Jordan homomorphisms holds and can be applied directly under the stated ring hypotheses (unit in A, char(B) > n).

What would settle it

A concrete pair of rings A with a unit and B with characteristic greater than n that admits an n-Jordan homomorphism which is neither an n-homomorphism nor an anti-n-homomorphism, while every Jordan homomorphism remains a homomorphism or anti-homomorphism, would falsify the claim.

read the original abstract

By using a variation of a theorem on $n$-Jordan homomorphisms due to Herstein, we deduce the following G. An's result: Let $ A $ and $ B $ be two rings where $ A $ has a unit and $ char(B)> n. $ If every Jordan homomorphism from $ A $ into $ B $ is a homomorphism (anti-homomorphism), then every $n$-Jordan homomorphism from $ A $ into $ B $ is an $n$-homomorphism (anti-$n$-homomorphism).

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 is a short note that re-derives An's known result on n-Jordan homomorphisms by applying a variation of Herstein's theorem, without adding new theorems or frameworks.

read the letter

The main takeaway is that the central claim is not new. The paper states up front that it deduces G. An's result: if every Jordan homomorphism from unital A to B (char B > n) is a homomorphism or anti-homomorphism, then every n-Jordan homomorphism is an n-homomorphism or anti-n-homomorphism. It reaches this by invoking a variation of Herstein's theorem on n-Jordan maps and applying it directly under the given hypotheses. The argument is presented as a clean logical step rather than a first-principles derivation. What the paper does well is keep the deduction short and explicit, with the ring conditions stated plainly and the link between the Jordan and n-Jordan cases made without extra notation or machinery. The citation pattern is straightforward, pointing to Herstein and An without circularity or self-reference issues. The soft spots are proportionate to the scope. The result itself was already obtained by An, so the contribution reduces to the alternative route through the Herstein variation. If that variation is merely restated rather than proved or strengthened here, the note functions more as an exposition than an advance. The abstract does not claim the variation is original, which keeps expectations in check. No load-bearing gaps appear in the structure described, and the hypotheses align with standard ring-theory settings. This paper is for specialists already following the Jordan homomorphism literature who want a compact alternative proof for comparison. A reader new to the area would get little from it. It deserves a serious referee because the logic is tight, the hypotheses are standard, and short clarifying notes can still be worth archiving even when they do not reorganize the field. I would send it out rather than desk-reject.

Referee Report

0 major / 1 minor

Summary. The manuscript is a short note that invokes a variation of Herstein's theorem on n-Jordan homomorphisms to deduce G. An's result: Let A and B be rings with A unital and char(B) > n. If every Jordan homomorphism A → B is a homomorphism (resp. anti-homomorphism), then every n-Jordan homomorphism A → B is an n-homomorphism (resp. anti-n-homomorphism).

Significance. If the invoked variation of Herstein's theorem holds under the stated hypotheses, the note supplies a direct, concise deduction of An's implication. This may streamline proofs in the literature on generalized ring homomorphisms and clarify the relationship between Jordan and n-Jordan maps without additional assumptions.

minor comments (1)

The abstract refers to 'a variation of a theorem on n-Jordan homomorphisms due to Herstein' but does not restate the precise statement of the variation used; including a one-sentence formulation of the variation (with citation) would make the deduction self-contained for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our note and for the positive assessment. We appreciate the recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript is a short conditional deduction that invokes an explicitly cited external variation of Herstein's theorem on n-Jordan homomorphisms and applies it directly under the stated hypotheses (A unital, char(B) > n) to recover the target implication for n-Jordan maps. No step reduces a claim to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain; the variation is referenced rather than reproved or assumed as prior work by the present author, so the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests only on standard ring axioms, the definition of Jordan and n-Jordan maps, and a cited variation of Herstein's theorem; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (1)

standard math Standard axioms of associative rings together with the assumption that A contains a multiplicative unit and char(B) > n
Invoked to ensure the algebraic identities used in the variation of Herstein's theorem are well-defined and invertible where needed.

pith-pipeline@v0.9.0 · 5369 in / 1224 out tokens · 50413 ms · 2026-05-10T05:07:33.200292+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references

[1]

An, Characterizations ofn-Jordan homomorphisms, Linear and Multi- linear Algebra, 66(4)(2018), 671-680

G. An, Characterizations ofn-Jordan homomorphisms, Linear and Multi- linear Algebra, 66(4)(2018), 671-680

2018
[2]

Bodaghi and H

A. Bodaghi and H. Inceboz,n-Jordan homomorphisms on commutative algebras, Acta Math. Univ. Comenianae, 87(1)(2018), 141-146

2018
[3]

Gselmann, On approximaten-Jordan homomorphisms, Annales Mathe- maticae Silesianae, 28(2014), 47-58

E. Gselmann, On approximaten-Jordan homomorphisms, Annales Mathe- maticae Silesianae, 28(2014), 47-58

2014
[4]

I. N. Herstein, Jordan homomorphisms, Trans. Amer. Math. Soc., 81(2)(1956), 331-341

1956
[5]

Ecole Normale Sup´ erieure Avenue Oued Akreuch Takaddoum, BP 5118, Rabat Morocco E-mail: mohammed.elazhari@yahoo.fr 3

Yang-Hi Lee, Stability ofn-Jordan homomorphisms from a normed algebra to a Banach algebra, Abstract and Applied Analysis, 2013(2013), Article ID 691025, 5 pages. Ecole Normale Sup´ erieure Avenue Oued Akreuch Takaddoum, BP 5118, Rabat Morocco E-mail: mohammed.elazhari@yahoo.fr 3

2013