理想在环中的作用

Function of Ideal in Ring

摘要: 理想在环中的作用与不变子群在群论中的作用相仿。不变子群在群论中起着非常重要的作用，是群的同态与同构相关结论获得的前提和基础。本文通过类比不变子群在群论中的作用，分析了理想对刻画环的性质所起的重要作用，阐明了理想的性质在决定环的同态与同构关系中的重要性，进一步说明了如何利用理想来构造有限域，进而建立数域上的同态及同构关系。

Abstract: The role of ideal in ring is similar to the invariant subgroup in group theory. Invariant subgroup plays a very important role in group theory, and is the premise and basis for the conclusion of group homomorphism and isomorphism. This paper analyzes the important role of ideals in characterizing the properties of rings by means of the function of analogous invariant subgroups in group theory, expounds the importance of the properties of ideals in determining the homomorphism and isomorphism of rings, and further illustrates how to construct finite fields by means of ideals, and then establish the homomorphism and isomorphism relations over number fields.