Filtered noetherian rings having homological finiteness

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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.