$(\sigma ,\tau )$-Derivations of Number Rings with Coding Theory Applications

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.