Co-Noether环和Co-Artin环

首先把P.V’amos在The dual of the notion of“finitely generated”中给出的R-范畴中的“有限嵌入”和V·A·Hiremath在Cofinitely generated and confi-nitely related modules中给出的“E(S)-余有限生成”两个概念加以统一并简称为“余有限生成”。通过“有限生成”和“余有限生成”两个概念给出Noether环和Artin环的系统刻划并通过对偶引入余Noether环和余Artin环。并在交换环的情况下讨论了余Noether环与余Artin环之间的关系及它们在一个极大理想上的局部化的性质:如果一个环R是Artin环,则R是余Neother环当且仅当它是余Artin环。我们也看到,在交换环时,余Noether环和余Artin分别是Noether环和Artin环的推广。

On Co-Noetherian and Co-Artinian Rings

We first establish the equivalence of two concepts, finite embeddedness in the category of R-modules (gilen by P.Va'mos in The Dual of the Notion of"Finitely Generated") and E(S)-finite cogeneration (given by V.A Hiremath inCofinitely Generated and Cofinitely Related Modules) and call it finite co-gene-ration of modules. A systematical characterization of Noetherian and Artinianrings is given using the two concepts of finite generation and finite co-genera-tion of modules, while Co-Noetherian and Co-Artian rings are introduced bydualisation. The interrelation between co-Noetherian and co-Artinain rings aswell as their relation to the localization at a maximal ideal in commutativecase are studied. If the ring R is Artinian, then R is co-Noetherian if and onlyif it is co-Artinian. We also see that in commutative case co-Noetherian andco-Artinian rings are generalizations of Noetherian and Artinian rings.