整数环的性质与应用研究

Research on Properties and Applications of Integer Rings

摘要: 整数环作为近世代数中的一个重要代数结构，其相关研究在群、环、域中非常重要。然而，整数环的构造、性质及其应用，有待于进一步的研究。本文主要给出整数环的构造，证明其基本性质。同时，研究整数环的判定、整数环与理想以及其他环类的联系、代数整数环上Ramanujan展开的应用。

Abstract: As an important algebraic structure in modern algebra, integer rings are very important in groups, rings, and fields. However, the structure, properties and applications of integer rings await further study. This article mainly gives the construction of integer rings and proves their basic properties. At the same time, the determination of integer rings, the relationship between integer rings and ideals and other ring types, and the application of Ramanujan expansion on algebraic integer rings are studied.