戴德金对理想论的贡献

Dedekind's Ideal Theory

理想概念源于理想数,是抽象代数学特别是交换环理论最核心的概念之一,经过一百多年的发展形成了较为完善的抽象理想论,很多数学家都参与其中,戴德金就是其中一位关键人物,他用理想代替了理想数,也就是由数过渡到集合,实现了早期代数数论的一般化。我们在搜集整理相关文献的基础上,通过对概念、方法和背景的考察与分析,以及通过与同时代数学家克罗内克的比较研究,阐释戴德金对理想论的贡献。

Ideal came from ideal number. It is one of the most important concepts in abstract algebra, especially the theory of commutative ring. The ideal theory took more than one hundred years to be mature. Many mathematicians contributed to it, among whom Dedekind was a key figure who stated the ideal concept and developed it from number to set, thus making the early algebraic number theory systematic. By document researching, concept analyzing and comparing him with Kronecker, we review Dedekind's contribution to the ideal theory.