({σ}, {τ})-Derivations of Number Rings with Coding Theory Applications
Pith reviewed 2026-05-23 07:42 UTC · model grok-4.3
The pith
All (σ, τ)-derivations of algebraic integers in quadratic fields are characterized with explicit bases.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We characterize all (σ, τ)-derivations and inner (σ, τ)-derivations of the ring of algebraic integers of a quadratic number field, determine the rank and an explicit basis of the derivation algebra, solve the twisted derivation problem in this setting, and give parallel characterizations for the ring of integers in a p-th cyclotomic field and in a bi-quadratic ring Z[√m, √n], conjecturing an if-and-only-if criterion for inner derivations in the cyclotomic case; as a consequence we construct Hom-IDD codes.
What carries the argument
The (σ, τ)-Leibniz rule, which requires a map D on the ring to satisfy D(ab) = σ(a)D(b) + D(a)τ(b) for given ring automorphisms σ and τ.
If this is right
- The derivation algebra on quadratic integer rings has determined rank and explicit basis.
- The twisted derivation problem is solved for quadratic and bi-quadratic rings.
- Hom-IDD codes arise directly from the derived bases in each case.
- An if-and-only-if condition is conjectured for inner (σ, τ)-derivations on cyclotomic rings.
Where Pith is reading between the lines
- The direct-computation method via explicit arithmetic could be tested on other families of number rings whose integral bases are known.
- The resulting Hom-IDD codes supply a new family whose minimum distances and dimensions can be compared with classical constructions.
- Solutions of the twisted derivation problem in these commutative cases suggest analogous questions for non-commutative orders or higher-degree extensions.
Load-bearing premise
The rings possess explicit Z-bases and arithmetic allowing every candidate map to be found by solving systems of linear equations.
What would settle it
A single map D on Z[√d] that satisfies the (σ, τ)-Leibniz rule for some automorphisms σ, τ yet lies outside the linear span of the stated basis would falsify the characterization.
read the original abstract
In this article, we study $(\sigma, \tau)$-derivations of number rings by considering them as commutative unital $\mathbb{Z}$-algebras. We begin by characterizing all $(\sigma, \tau)$-derivations and inner $(\sigma, \tau)$-derivations of the ring of algebraic integers of a quadratic number field. Then we characterize all $(\sigma, \tau)$-derivations of the ring of algebraic integers $\mathbb{Z}[\zeta]$ of a $p^{\text{th}}$-cyclotomic number field $\mathbb{Q}(\zeta)$ ($p$ odd rational prime and $\zeta$ a primitive $p^{\text{th}}$-root of unity). We also conjecture (using SageMath and MATLAB) an \enquote{if and only if} condition for a $(\sigma, \tau)$-derivation $D$ on $\mathbb{Z}[\zeta]$ to be inner. We further characterize all $(\sigma, \tau)$-derivations and inner $(\sigma, \tau)$-derivations of the bi-quadratic number ring $\mathbb{Z}[\sqrt{m}, \sqrt{n}]$ ($m$, $n$ distinct square-free rational integers). In each of the above cases, we also determine the rank and an explicit basis of the derivation algebra consisting of all $(\sigma, \tau)$-derivations of the number ring. As a consequence, we solve the twisted derivation problem in the ring of algebraic integers of a quadratic number field and in a bi-quadratic number ring, and we conjecture a solution of the twisted derivation problem in the ring of algebraic integers of a $p^{\text{th}}$-cyclotomic number field. Finally, we give the applications of our work in coding theory by constructing Hom-IDD codes.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper characterizes all (σ, τ)-derivations of the rings of algebraic integers in quadratic number fields, odd-prime cyclotomic fields ℤ[ζ_p], and biquadratic rings ℤ[√m, √n] (m, n distinct square-free), treating them as commutative unital ℤ-algebras. It determines the rank and gives explicit bases for the derivation algebra in each case, solves the twisted derivation problem for the quadratic and biquadratic rings, conjectures an if-and-only-if criterion for inner derivations in the cyclotomic case on the basis of SageMath/MATLAB computations, and constructs Hom-IDD codes as an application.
Significance. The explicit algebraic characterizations and bases supply concrete, computable descriptions of the (σ, τ)-derivation modules for these low-rank number rings. Because the rings are free ℤ-modules of small rank whose minimal polynomials are known, the (σ, τ)-Leibniz conditions reduce to finite linear systems whose solutions yield the stated ranks and bases; this approach is internally consistent and directly supports the claimed solutions of the twisted derivation problem in two of the three settings. The coding-theory constructions illustrate a potential downstream use.
minor comments (3)
- Abstract: the phrase “we also conjecture (using SageMath and MATLAB) an ‘if and only if’ condition” should be accompanied, in the body, by the precise conjectured criterion and the range of primes p for which the computational evidence was obtained.
- The term “Hom-IDD codes” appears in the abstract and conclusion without definition or reference; a short explanation or citation to the relevant coding-theory literature is needed for readers outside that subfield.
- Notation: the symbols σ and τ are introduced as ring endomorphisms but their precise domains and images (e.g., whether they fix ℤ pointwise) should be stated uniformly in the opening paragraphs of each case study.
Simulated Author's Rebuttal
We thank the referee for the careful reading and positive assessment of the manuscript, including the accurate summary of our results on (σ, τ)-derivations for quadratic, cyclotomic, and biquadratic rings, the solution of the twisted derivation problem in two cases, the conjecture in the cyclotomic case, and the coding-theory application. We note the recommendation for minor revision.
Circularity Check
No circularity; characterizations obtained via direct solution of linear systems on explicit Z-bases
full rationale
The paper determines all (σ, τ)-derivations by imposing the twisted Leibniz rule on the standard integral bases of quadratic, biquadratic, and cyclotomic rings. These rings are free Z-modules of small finite rank whose minimal polynomials are known explicitly; the derivation conditions therefore reduce to finite systems of linear equations over the ring. Solving those systems yields the stated ranks and bases without invoking fitted parameters, self-citation chains, or external uniqueness theorems. The cyclotomic inner-derivation statement is presented as a computational conjecture rather than a proved claim, and the coding-theory application is a downstream construction that does not feed back into the algebraic results. No step in the derivation chain is definitionally equivalent to its input or statistically forced by a prior fit.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The rings of algebraic integers are commutative unital Z-algebras
Reference graph
Works this paper leans on
-
[1]
On ( σ, τ)-derivations of group algebra as category characters
Aleksandr Alekseev, Andronick Arutyunov, and Sergei Silvestr ov. On ( σ, τ)-derivations of group algebra as category characters. arXiv preprint arXiv:2008.00390, 2020
-
[2]
New types of permuting n-derivations with their applications on associative rings
Mehsin Jabel Atteya. New types of permuting n-derivations with their applications on associative rings. Symmetry, 12(1):46, 2019
work page 2019
-
[3]
M. Bresar. On the composition of ( α, β)-derivations of rings and applications to von Neumann algebras. Acta Sct. Math , 56:369–375, 1992. 33
work page 1992
-
[4]
On generalized ( α, β)-derivations on lattices
Muhammad Anwar Chaudhry and Zafar Ullah. On generalized ( α, β)-derivations on lattices. Quaestiones Mathematicae , 34(4):417–424, 2011
work page 2011
-
[5]
A note on ( σ, τ)-derivations on commutative algebras
Dishari Chaudhuri. A note on ( σ, τ)-derivations on commutative algebras. arXiv preprint arXiv:1912.12812, 2019
-
[6]
Derivations on group algebras w ith coding theory applications
Leo Creedon and Kieran Hughes. Derivations on group algebras w ith coding theory applications. Finite Fields and Their Applications , 56:247–265, 2019
work page 2019
-
[7]
Linear codes using skew polynomia ls with auto- morphisms and derivations
Boucher Delphine and Felix Ulmer. Linear codes using skew polynomia ls with auto- morphisms and derivations. Designs, codes and cryptography , 70:405–431, 2014
work page 2014
-
[8]
M. Haetinger, C. Ashraf and S. Ali. On higher derivations: a surve y. Int. J. Math., Game Theory and Algebra , 18:359–379, 2011
work page 2011
- [9]
-
[10]
Kamali Ardakani and Bijan Davvaz
L. Kamali Ardakani and Bijan Davvaz. f-derivations and (f,g)- derivations of MV- algebras. Journal of Algebraic Systems , 1(1):11–31, 2013
work page 2013
-
[11]
On (f,g)-derivations of incline algebras
Kyung Ho Kim. On (f,g)-derivations of incline algebras. Journal of the Chungcheong Mathematical Society, 27(4):643–649, 2014
work page 2014
-
[12]
Unbounded derivations in algebr as associated with monothetic groups
Slawomir Klimek and Matt McBride. Unbounded derivations in algebr as associated with monothetic groups. Journal of the Australian Mathematical Society , 111(3):345– 371, 2021
work page 2021
-
[13]
Daniel A. Marcus. Number fields . Universitext. Springer, 2nd edition, 2018
work page 2018
-
[14]
On derivatio ns in rings and their applications
Ashraf Mohammad, Ali Shakir, and Haetinger Claus. On derivatio ns in rings and their applications. The Aligarh Bull of Math , 25(2):79–107, 2006
work page 2006
-
[15]
On ( α, β)-derivations in BCI-algebras
G Muhiuddin and Abdullah M Al-Roqi. On ( α, β)-derivations in BCI-algebras. Discrete Dynamics in Nature and Society , 2012:11, 2012
work page 2012
-
[16]
Generalized (f, g)-derivations of lattices
AS ¸CI Mustafa and S ¸ahin Ceran. Generalized (f, g)-derivations of lattices. Mathematical Sciences and Applications E-Notes , 1(2):56–62, 2013
work page 2013
-
[17]
On prime and semiprime ring s with gen- eralized derivations and non-commutative Banach algebras
Mohd Arif Raza and Nadeem Ur Rehman. On prime and semiprime ring s with gen- eralized derivations and non-commutative Banach algebras. Proceedings-Mathematical Sciences, 126(3):389–398, 2016
work page 2016
-
[18]
Algebraic Number Theory and Fermat’s Last Theorem
Ian Stewart and David Tall. Algebraic Number Theory and Fermat’s Last Theorem . A. K. Peters, 3rd edition, 2002. 34
work page 2002
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.