交换半环的局部化

Localization of commutative semi-ring

环的全体素理想的集合称为环的素谱 .在环的素谱上可定义Zariski-拓扑 .非交换环的素谱按Zariski-拓扑不成谱空间 ,但可以构造它的谱闭包 .谱闭包恰是非交换环的理想半环的某种交换化半环的谱空间 .因而就引出了研究交换半环的素理想的课题 .为此研究交换半环在素理想处的局部化 .先构造交换半环关于其乘法封闭子集的分式半环 ;进而对半环及其上的半模引入局部化概念 ;证明了分式半模可用纯量的扩张 ,即张量积来表示 .

The set of all prime ideals of a ring is called the spectrum of the ring, which can be endowed with Zariski topology. The spectrum of a non commutative ring is not a spectral space under Zariski topology, but a spectral closure of the spectrum can be constructed. This spectral closure is exactly the spectrum of a commutative semi ring associated with the semi ring of ideals of the ring. So, the prime ideals of a commutative semi ring draw attention in the theory of non commutative ring. This paper considers the localization of a commutative semi ring at the prime ideal. First, semi ring of fractions of a semi ring about a multiplicative closed subset in it is constructed. Then, the notion of localization for the semi ring and the semi module are introduced. Furthermore, the semi module of fractions is represented as the extension of scalars, which is defined by a tensor product.