2-Local and local derivations on Jordan matrix rings over commutative involutive rings

F.N. Arzikulov; N.M. Umrzaqov; O.O. Nuriddinov; Sh.A. Ayupov

arxiv: 2108.03993 · v3 · submitted 2021-07-25 · 🧮 math.RA · math.OA

2-Local and local derivations on Jordan matrix rings over commutative involutive rings

Sh.A. Ayupov , F.N. Arzikulov , N.M. Umrzaqov , O.O. Nuriddinov This is my paper

Pith reviewed 2026-05-24 13:09 UTC · model grok-4.3

classification 🧮 math.RA math.OA

keywords Jordan ringderivation2-local derivationinner derivationspatial derivationself-adjoint matrixinvolutive ring

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The pith

Every 2-local inner derivation on the Jordan ring of self-adjoint matrices over a commutative involutive ring is a derivation.

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

The paper establishes that every 2-local inner derivation on the Jordan ring of self-adjoint matrices over a commutative involutive ring is in fact a derivation. This shows that a condition checked only on pairs of elements forces the map to satisfy the full derivation identity on the entire ring. The authors extend the argument to Jordan algebras consisting of infinite-dimensional self-adjoint matrix-valued maps, proving that 2-local spatial derivations are spatial derivations and that local spatial derivations are derivations. A sympathetic reader cares because the result reduces the work of verifying derivation properties to a strictly weaker local test in these nonassociative structures.

Core claim

We prove that every 2-local inner derivation on the Jordan ring of self-adjoint matrices over a commutative involutive ring is a derivation. We also apply our technique to various Jordan algebras of infinite dimensional self-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation. It is also proved that every local spatial derivation on the same Jordan algebras is a derivation.

What carries the argument

The Jordan ring of self-adjoint matrices over a commutative involutive ring, together with the reduction of 2-local inner derivations to ordinary derivations.

If this is right

Every 2-local inner derivation on the Jordan ring of self-adjoint matrices is a derivation.
Every 2-local spatial derivation on the infinite-dimensional matrix-valued Jordan algebras is a spatial derivation.
Every local spatial derivation on the same infinite-dimensional Jordan algebras is a derivation.

Where Pith is reading between the lines

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

The same locality-to-global reduction may hold for other families of Jordan algebras built from involutive rings.
The technique could be tested on derivations of Jordan structures over noncommutative base rings to see where commutativity is essential.
Practical verification of whether a map is a derivation on large matrix Jordan rings may be reduced to checking pairs of elements.

Load-bearing premise

The base ring is commutative and involutive so that the self-adjoint matrices carry a Jordan ring structure in which inner and spatial derivations are defined.

What would settle it

An explicit commutative involutive ring together with a 2-local inner derivation on its self-adjoint matrix Jordan ring that fails to satisfy the derivation identity on some triple of elements.

read the original abstract

In the present paper we prove that every 2-local inner derivation on the Jordan ring of self-adjoint matrices over a commutative involutive ring is a derivation. We also apply our technique to various Jordan algebras of infinite dimensional self-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation. It is also proved that every local spatial derivation on the same Jordan algebras is a 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

The paper shows that 2-local inner derivations on Jordan self-adjoint matrix rings over commutative involutive rings are derivations, with similar results for spatial derivations in some infinite-dimensional cases.

read the letter

The core result is that every 2-local inner derivation on the Jordan ring of self-adjoint matrices over a commutative involutive ring is actually a derivation. The authors extend the same approach to Jordan algebras of infinite-dimensional self-adjoint matrix-valued maps and show that 2-local spatial derivations are spatial and that local spatial derivations are derivations. This is new for these exact classes of rings and algebras. Earlier work on local derivations exists, but the specific statements here for Jordan matrix rings with the involution condition and the infinite-dimensional map versions do not appear in the referenced prior literature. The technique they use to move from local to global seems consistent across the finite and infinite settings, which is a clear strength. The proofs are presented as direct implications without obvious circularity or self-referential fitting. The commutativity and involution on the base ring are the standard setup needed to get the Jordan structure on the self-adjoint matrices, so that assumption is not a hidden weakness. The main limitation is the narrow scope. These are technical results in a specialized corner of non-associative algebra, unlikely to change thinking outside Jordan derivation theory or operator algebra applications. Without the full proofs in front of me it is impossible to spot any gaps in the arguments, but the abstract gives no sign of load-bearing flaws or invented entities. This paper is for people already working on derivations in Jordan algebras or related ring-theoretic questions. A reader who follows that literature would get a useful reference point from the specific theorems. It is worth sending to peer review so that experts can check the details of the proofs.

Referee Report

0 major / 1 minor

Summary. The paper proves that every 2-local inner derivation on the Jordan ring of self-adjoint matrices over a commutative involutive ring is a derivation. It applies the same technique to Jordan algebras consisting of infinite-dimensional self-adjoint matrix-valued maps on a set, showing that every 2-local spatial derivation is a spatial derivation and that every local spatial derivation on these algebras is a derivation.

Significance. If the proofs are correct, the results extend known facts about local and 2-local derivations from associative algebras to the Jordan setting, providing a uniform technique for handling inner and spatial derivations on matrix constructions over involutive rings. This could serve as a template for analogous questions in other nonassociative algebras.

minor comments (1)

The abstract states the existence of proofs but supplies no outline of the key steps or the precise definition of 'inner' versus 'spatial' derivations used in the Jordan context; a brief indication of the method in the abstract would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and for providing a concise summary of our results. The referee's assessment of the potential significance is appreciated. No specific major comments or criticisms were listed in the report, so we have no individual points requiring a detailed response at this time.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper states direct proofs that 2-local inner derivations are derivations and extends the technique to spatial derivations on Jordan algebras of matrix-valued maps. No equations or steps in the provided abstract reduce a claimed result to a fitted input, self-definition, or load-bearing self-citation by construction. The commutative involutive ring assumption is a standard enabling condition for the Jordan structure rather than a circular premise. This is the expected non-finding for a direct algebraic proof paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard axioms of Jordan algebras, derivations, and involutive rings; no free parameters or invented entities introduced.

axioms (1)

standard math Jordan algebra axioms and derivation properties over commutative involutive rings
Background assumptions standard in ring theory invoked to define the structures.

pith-pipeline@v0.9.0 · 5614 in / 994 out tokens · 23950 ms · 2026-05-24T13:09:40.342717+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · 1 internal anchor

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Arzikulov, Inﬁnite order decompositions of C*-algebras, Sp ringerPlus

F.N. Arzikulov, Inﬁnite order decompositions of C*-algebras, Sp ringerPlus. 5(1) (2016) 1–13

work page 2016
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Ayupov, F

Sh. Ayupov, F. Arzikulov, 2-Local derivations on associative an d Jordan matrix rings over involutive commutative rings, Linear Algebra Appl. 522 (2017) 28–50

work page 2017
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20(1) (2018) 38 –49

Ayupov Sh.A., Arzikulov F.N., 2-Local derivations on algebras of ma trix-valued functions on a compact, Vladikavkaz Mathematical journal. 20(1) (2018) 38 –49

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Sh. A. Ayupov, K. K. Kudaybergenov, A. M. Peralta, A survey o n local and 2-local deriva- tions on C ∗- and von Neumann algebras, in Topics in Functional Analysis and Algeb ra, Contemporary Mathematics, vol. 672, Amer. Math. Soc., Providen ce, RI, 2016, pp. 73–126

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Johnson, Local derivations on C*-algebras are derivations, Trans

B. Johnson, Local derivations on C*-algebras are derivations, Trans. Amer. Math. Soc. 353(2001), 313–325

work page 2001
[6]

Kadison, Local derivations, J

R. Kadison, Local derivations, J. Algebra 130(1990), 494–509

work page 1990
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Larson, A

D. Larson, A. Sourour, Local derivations and local automorph isms, Proc. Sympos. Pure Math. 51(1990), 187–194

work page 1990
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Weak 2-local derivations on $\mathbb{M}_n$

M. Niazi, A.M. Peralta, Weak-2-local derivations on Mn, FILOMAT (2017), available from http://arXiv/abs/1503.01346/

work page internal anchor Pith review Pith/arXiv arXiv 2017
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ˇSemrl, Local automorphisms and derivations on B(H), Proc

P. ˇSemrl, Local automorphisms and derivations on B(H), Proc. Amer. Math. Soc. 125 (1997) 2677–2680

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H. Upmeier, Derivations on Jordan C ∗-algebras, Math. Scand. 46 (1980) 251–264. 1 V.I.Romanovskiy Institute of Mathematics Uzbekistan Acad emy of Sciences, Tashkent, Uzbekistan 22 SH.A. AYUPOV, F.N. ARZIKULOV, N. M. UMRZAQOV, AND O. O. NUR IDDINOV 2 National University of Uzbekistan, Tashkent, Uzbekistan Email address : shavkat.ayupov@mathinst.uz 3 V.I. R...

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Arzikulov, Inﬁnite order decompositions of C*-algebras, Sp ringerPlus

F.N. Arzikulov, Inﬁnite order decompositions of C*-algebras, Sp ringerPlus. 5(1) (2016) 1–13

work page 2016

[2] [2]

Ayupov, F

Sh. Ayupov, F. Arzikulov, 2-Local derivations on associative an d Jordan matrix rings over involutive commutative rings, Linear Algebra Appl. 522 (2017) 28–50

work page 2017

[3] [3]

20(1) (2018) 38 –49

Ayupov Sh.A., Arzikulov F.N., 2-Local derivations on algebras of ma trix-valued functions on a compact, Vladikavkaz Mathematical journal. 20(1) (2018) 38 –49

work page 2018

[4] [4]

Sh. A. Ayupov, K. K. Kudaybergenov, A. M. Peralta, A survey o n local and 2-local deriva- tions on C ∗- and von Neumann algebras, in Topics in Functional Analysis and Algeb ra, Contemporary Mathematics, vol. 672, Amer. Math. Soc., Providen ce, RI, 2016, pp. 73–126

work page 2016

[5] [5]

Johnson, Local derivations on C*-algebras are derivations, Trans

B. Johnson, Local derivations on C*-algebras are derivations, Trans. Amer. Math. Soc. 353(2001), 313–325

work page 2001

[6] [6]

Kadison, Local derivations, J

R. Kadison, Local derivations, J. Algebra 130(1990), 494–509

work page 1990

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Larson, A

D. Larson, A. Sourour, Local derivations and local automorph isms, Proc. Sympos. Pure Math. 51(1990), 187–194

work page 1990

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Weak 2-local derivations on $\mathbb{M}_n$

M. Niazi, A.M. Peralta, Weak-2-local derivations on Mn, FILOMAT (2017), available from http://arXiv/abs/1503.01346/

work page internal anchor Pith review Pith/arXiv arXiv 2017

[9] [9]

ˇSemrl, Local automorphisms and derivations on B(H), Proc

P. ˇSemrl, Local automorphisms and derivations on B(H), Proc. Amer. Math. Soc. 125 (1997) 2677–2680

work page 1997

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Upmeier, Derivations on Jordan C ∗-algebras, Math

H. Upmeier, Derivations on Jordan C ∗-algebras, Math. Scand. 46 (1980) 251–264. 1 V.I.Romanovskiy Institute of Mathematics Uzbekistan Acad emy of Sciences, Tashkent, Uzbekistan 22 SH.A. AYUPOV, F.N. ARZIKULOV, N. M. UMRZAQOV, AND O. O. NUR IDDINOV 2 National University of Uzbekistan, Tashkent, Uzbekistan Email address : shavkat.ayupov@mathinst.uz 3 V.I. R...

work page 1980