Artin环局部化结构与一类商环的等价刻画

Equivalent Characterization of Localization Structure of Artin Rings and A Class of Quotient Rings

设R是Artin交换环,m_(i)(1≤i≤k)是R的k个极大理想.通过Artin环的Jacobson根幂零和中国剩余定理得到R到⊕R/m_(i)^(l)的同态是一种同构关系.进一步把Artin环写成有限个局部环Re_(i)的直和,发现这些直和加项与商环R/m_(i)^(l)是同构关系,并且与局部环R_(m_(i))的也是同构关系.

Let R be an Artin commutative ring,and m_(i)(1≤i≤k)be the k maximal ideals of R.Through the Jacobson root nilpotent of Artin ring and chinese remainder theorem,it is obtained that the homomorphism from R to ⊕R/m_(i)^(l)is an isomorphism.Further,Artin rings are written as the direct sum of finite local rings Re i.It is found that each addition of direct sum is isomorphic to a quotient ring R/m_(i)^(l),and it is isomorphic with local ring R_(m_(i)).