Auslander. Zariskian Ring Mold

The modules over Auslander regular or Auslander-Gorenstein rings were studied a lot during past few years. Now it is possible to re-discover those results on special modules over arbitrary Noetherian rings. Namely, those modules which satisfy the Auslander condition and their Ext modules also satisfy the Auslander condition. When considering Zariskian rings and their associate graded rings, it was found out that Auslander property on those special graded modules can be lifted to the original ones even without the hypothesis that the ring must have finite injective dimension. Some properties of Strongly Pure modules are also discussed.