Arithmetic in group rings and study of zero-sum problems in combinatorial number theory

一个环是一个代数系统，有两个操作：加法和乘法。例如，所有具有规则加法和乘法的整数形成一个环。 交换环K上的群G的群环是所有形式有限和的环KG。这是一个有吸引力的和重要的研究对象在代数。在这里群论，环理论，交换代数，代表理论和数论走到一起，在一个富有成效的方式。 我最近的工作揭示了群环及其单位群的代数结构（和内部算术），它在编码理论和组合数论中有重要的应用。这将是我未来5年研究计划的主要方向之一。我将研究关于群环的几个开放问题和问题，包括第一个Zassenhaus猜想-一个长期存在的研究问题-和正规化问题。我也将研究星清洁和可逆群环的结构。 零和理论是组合数论的一个新的分支。最近我已经开始探索一些令人兴奋的创新联系组合数论和我的工作在群环。高卫东博士和我能够使用群环作为主要工具来研究零和理论中的几个公开问题，涉及几个重要的不变量，如达文波特常数和序列的指数。我们已经在这一研究领域做出了重大贡献。进一步的研究正在进行中，我们预计在不久的将来报告重大进展。我还参与了其他领域，如（组合）群论，我已经获得了新的成果，有关团体与小平方财产，和代数编码理论。 我的研究成果不仅将增加知识的领域群环和零和理论，但也激励其他研究人员在这些领域和其他领域，我的研究可能适用。

A ring is an algebraic system with two operations: addition and multiplication. For example, all integers with regular addition and multiplication form a ring. The group ring of a group G over a commutative ring K is the ring KG of all formal finite sums. It is an attractive and important object of study in algebra. Here group theory, ring theory, commutative algebra, representation theory and number theory come together in a fruitful way. My recent work has shed light on the algebraic structures (and the internal arithmetic ) of group rings and their unit groups, and it has important applications in coding theory and combinatorial number theory. This will continue to be one of the main streams of my research program for the next 5 years. I will study several open problems and conjectures regarding group rings, including the first Zassenhaus Conjecture - a long standing research problem- and the normalizer problem. I will also investigate the structure of star-clean and reversible group rings. Zero-sum theory is a new branch of combinatorial number theory. Recently I have begun exploring some exciting innovative connections between combinatorial number theory and my work in group rings. Dr. Weidong Gao and I are able to use group rings as the main tool to investigate several open problems in zero-sum theory concerning several important invariants such as the Davenport constant and the index of sequences. We have already made significant contributions in this research area. Further research is ongoing, and we anticipate reporting significant advances in the immediate future. I am also involved in other areas such as (combinatorial) group theory where I have obtained new results concerning groups with small squaring property, and algebraic coding theory. The outcomes of my research will not only add knowledge to the areas of group rings and zero-sum theory but also inspire other researchers in these areas and other areas to which my research may apply.