Higher power polyadic group rings
Steven Duplij
本文引入并系统发展了多元群环理论，这是经典群环ℛ[𝖦]的高元数推广。我们构建了这些结构的基本运算，定义了由(m_r,n_r)-环和n_g元群构建的多元群环R^[𝐦_r,𝐧_r]=ℛ^[m_r,n_r][𝖦^[n_g]]的𝐦_r元加法和𝐧_r元乘法。一个核心结果是推导了这些元数之间相互关联的"量子化"条件，这些条件受元数自由原则支配，该原则也扩展到具有高次多元幂的运算。我们建立了关键的代数性质，包括完全结合性的条件以及零元和单位元的存在性。多元增广映射和增广理想的概念得到了推广，为经典理论提供了桥梁。该框架通过具体示例进行了说明，巩固了理论构造。这项工作为环论建立了新的基础，在密码学和编码理论中具有潜在应用，正如最近利用多元结构的方案所证明的那样。
This paper introduces and systematically develops the theory of polyadic group rings, a higher arity generalization of classical group rings ℛ[𝖦]. We construct the fundamental operations of these structures, defining the 𝐦_r-ary addition and 𝐧_r-ary multiplication for a polyadic group ring R^[𝐦 _r,𝐧_r]=ℛ^[m_r,n_r][𝖦^[n_g]] built from a nonderived (m_r,n_r)-ring and a nonderived n_g-ary group. A central result is the derivation of the "quantization" conditions that interrelate these arities...