2024/12/04 by Praveen Manju, Manju, Praveen, Sharma, Rajendra Kumar
Mathematics · #11C20 #11R04 #11R11 #11R16 #11R18 #13N15 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2412.03500
openalex publication_date 2024/12/04 · openalex created_date 2024/12/06 · openalex updated_date 2026/07/28
In this article, we study (σ, τ)-derivations of number rings by considering them as commutative unital ℤ-algebras. We begin by characterizing all (σ, τ)-derivations and inner (σ, τ)-derivations of the ring of algebraic integers of a quadratic number field. Then we characterize all (σ, τ)-derivations of the ring of algebraic integers ℤ[ζ] of a pth-cyclotomic number field ℚ(ζ) (p odd rational prime and ζ a primitive pth-root of unity). We also conjecture (using SageMath and MATLAB) an \enquoteif and only if condition for a (σ, τ)-derivation D on ℤ[ζ] to be inner. We further characterize all (σ, τ)-derivations and inner (σ, τ)-derivations of the bi-quadratic number ring ℤ[√(m), √(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 (σ, τ)-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 pth-cyclotomic number field. Finally, we give the applications of our work in coding theory by constructing Hom-IDD codes.