$(σ, τ)$-Derivations of Number Rings with Coding Theory Applications
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| Format: | Preprint |
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2024
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| _version_ | 1866908933862457344 |
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| author | Manju, Praveen Sharma, Rajendra Kumar |
| author_facet | Manju, Praveen Sharma, Rajendra Kumar |
| contents | In this article, we study $(σ, τ)$-derivations of number rings by considering them as commutative unital $\mathbb{Z}$-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 $\mathbb{Z}[ζ]$ of a $p^{\text{th}}$-cyclotomic number field $\mathbb{Q}(ζ)$ ($p$ odd rational prime and $ζ$ a primitive $p^{\text{th}}$-root of unity). We also conjecture (using SageMath and MATLAB) an \enquote{if and only if} condition for a $(σ, τ)$-derivation $D$ on $\mathbb{Z}[ζ]$ to be inner. We further characterize all $(σ, τ)$-derivations and inner $(σ, τ)$-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 $(σ, τ)$-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 $p^{\text{th}}$-cyclotomic number field. Finally, we give the applications of our work in coding theory by constructing Hom-IDD codes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_03500 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $(σ, τ)$-Derivations of Number Rings with Coding Theory Applications Manju, Praveen Sharma, Rajendra Kumar Number Theory Commutative Algebra Rings and Algebras 13N15, 11R04, 94B05, 11R11, 11R18, 11R16, 11C20 In this article, we study $(σ, τ)$-derivations of number rings by considering them as commutative unital $\mathbb{Z}$-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 $\mathbb{Z}[ζ]$ of a $p^{\text{th}}$-cyclotomic number field $\mathbb{Q}(ζ)$ ($p$ odd rational prime and $ζ$ a primitive $p^{\text{th}}$-root of unity). We also conjecture (using SageMath and MATLAB) an \enquote{if and only if} condition for a $(σ, τ)$-derivation $D$ on $\mathbb{Z}[ζ]$ to be inner. We further characterize all $(σ, τ)$-derivations and inner $(σ, τ)$-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 $(σ, τ)$-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 $p^{\text{th}}$-cyclotomic number field. Finally, we give the applications of our work in coding theory by constructing Hom-IDD codes. |
| title | $(σ, τ)$-Derivations of Number Rings with Coding Theory Applications |
| topic | Number Theory Commutative Algebra Rings and Algebras 13N15, 11R04, 94B05, 11R11, 11R18, 11R16, 11C20 |
| url | https://arxiv.org/abs/2412.03500 |