Gorenstein dimension and Gorenstein algebras

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We consider noncommutative filtered rings and filtered modules over them. Our main objective is to calculate homological invariants and some equalities or inequalities of homological invariants, like Gorenstein dimension, grade. When this is the case, we see that associated graded rings and modules play important role to handle such homological invariants. We get that Gorenstein dimension of an associated graded module grM is greater than or equal to that of a filtered module M over a filtered ring. We also get that, under general condition as before, grade of an associated graded module coincides with that of a filtered module. This condition is described by module-wise condition. Hence we can study grade of modules over not necessarily Gorenstein filtered rings and see that there is a duality given by Ext-group between left and right Cohen-Macaulay modules over such filtered ring. The properties of Gorenstein filtered rings are also studied under the point of view of generalization of properties of Auslander regular ring. We get, for example, that a module is pure if and only if it is geometrically pure and has no isolated associated primes.