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

环是一个代数系统，有两种运算：加法和乘法。例如，所有具有规则加法和乘法的整数构成一个环。交换环K上群G的群环是所有形式有限和的环KG。它是代数中一个有吸引力的重要研究对象。在这里，群论、环论、交换代数、表示论和数论以一种富有成效的方式结合在一起。******我最近的工作揭示了群环及其单位群的代数结构（和内部算法），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年的主要研究方向之一。我将研究几个关于群环的开放问题和猜想，包括第一个Zassenhaus猜想——一个长期存在的研究问题——和规范化问题。我还将研究星净和可逆群环的结构。******零和理论是组合数论的一个新分支。最近，我开始探索组合数论与我在群环中的工作之间一些令人兴奋的创新联系。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开放问题，这些问题涉及到达文波特常数和序列索引等几个重要的不变量。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会报告重大进展。我还参与了其他领域，如（组合）群论，在那里我获得了关于具有小平方性质的群的新结果，以及代数编码理论。******我的研究成果不仅将为群环和零和理论领域增加知识，而且还将激励这些领域的其他研究人员以及我的研究可能适用的其他领域。

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.