Filtered noetherian rings having homological finiteness

定义了由环的滤波器导出的滤波伪紧代数的拓扑结构。然后我们可以用代数方法来研究一个滤波的伪紧代数。通过定义由滤波器导出的滤波伪紧代数的拓扑，给出了滤波伪紧代数的纯代数定义。也就是说，假设滤波器是由极大态导出的，则滤波后的伪紧代数是诺etherian和半完美的。noether半完美环上所有有限生成模的范畴具有很好的性质，即其中的所有模都有一个射影覆盖。这一事实与近似理论密切相关。因此，滤波伪紧代数被期望用于非交换诺ether代数的表示理论。

We define the topology of a filtered pseudo-compact algebra induced from a filter of a ring. Then we can use algebraic method to study a filtered pseudo-compact algebra. By defining toplogy of filtered pseudo compact algebra induced from the filter, we give filtered pseudo compact algebra pury algebraic definition of filtered pseudo compact algebra. That is, suppose that the filter is induced from the maximum idal then filtered pseudo compact algebra is noetherian and semiperfect. The category of all finitely generated modules over noetherian semiperfect ring has good property, i.e. all the modules in it has a projective cover. This fact is highly connected to approximation theory. Hence filtered pseudo compact algebra is expected to the representation theory of non-commutative noetherian algebras.