联想类比在近世代数中的应用

The Application of Associative Analogy in Modern Algebra

摘要: 联想类比是数学中一个很重要的思想，在近世代数中有着深刻的应用，在群与环的定义里，在用模的同余关系将集合、群、环进行分类进而得到性质比较好的新分类的集合、剩余类加群、剩余类环里，在集合、群、环关于映射的规律里，联想类比思想都有着深刻的意义。不仅在近世代数内部，联想类比还可以应用于多个数学分支里，用同态去解释求导，将运动构成一个群。在不同的自然科学领域，联想类比依然有着充分的体现，晶体的对称性，亚原子粒子的对称性，场论都与群论有着密不可分的联系。

Abstract: Associative analogy is a very important idea in mathematics, which has a profound application in modern algebra. In the definition of group and ring, the congruence relation of modules is used to classify sets, groups and rings, and then new classified sets, residual classes plus groups and residual class rings with better properties are obtained. In the mapping rules of sets, groups and rings, associative analogy thoughts are profound significance. Not only in modern algebra, associative analogy can also be applied to many branches of mathematics. Homomorphism is used to explain and derive, and the motion is formed into a group. In different fields of natural science, associative analogy is still fully reflected. The symmetry of crystal, subatomic particle and field theory is closely related to group theory.