ar X iv : m at h / 04 09 16 3 v 2 [ m at h . R A ] 2 S ep 2 00 7 On the Grade of Modules over Noetherian Rings ∗ †

Let Λ be a left and right noetherian ring and modΛ the category of finitely generated left Λ-modules. In this paper we show the following results: (1) For a positive integer k, the condition that the subcategory of modΛ consisting of i-torsionfree modules coincides with the subcategory of modΛ consisting of i-syzygy modules for any 1 ≤ i ≤ k is left-right symmetric. (2) If Λ is an Auslander ring and N is in modΛ with gradeN = k < ∞, then N is pure of grade k if and only if N can be embedded into a finite direct sum of copies of the (k + 1)st term in a minimal injective resolution of Λ as a right Λ-module. (3) Assume that both the left and right self-injective dimensions of Λ are k. If grade ExtkΛ(M,Λ) ≥ k for any M ∈ modΛ and gradeExt i Λ(N,Λ) ≥ i for any N ∈ modΛ and 1 ≤ i ≤ k − 1, then the socle of the last term in a minimal injective resolution of Λ as a right Λ-module is non-zero.