Auslander-type Conditions

Throughout this article, Λ is a left and right Noetherian ring (unless stated otherwise), modΛ is the category of finitely generated left Λ-modules and 0 → Λ → I0(Λ) → I1(Λ) → · · · → Ii(Λ) → · · · is the minimal injective resolution of Λ as a left Λ-module. For a module M ∈ mod Λ and a non-negative integer n, recall that the grade of M , denoted by gradeM , is said to be at least n if ExtΛ(M, Λ) = 0 for any 0 ≤ i < n; and the strong grade of M , denoted by s.gradeM , is said to be at least n if gradeX ≥ n for any submodule X of M (see [3] and [7]). Bass in [8] proved the following result.