On the Grade of Modules over Noetherian Rings

Let be a left and right noetherian ring and mod the category of finitely generated left -modules. In this article, we show the following results. 1 For a positive integer k, the condition that the subcategory of mod consisting of i-torsionfree modules coincides with the subcategory of mod consisting of i-syzygy modules for any 1 ≤ i ≤ k is left-right symmetric. 2 If is an -Gorenstein ring and N is in mod op 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 as a right -module. 3 Assume that both the left and right self-injective dimensions of are k. If grade Ext M ≥ k for any M ∈ mod and grade Ext N ≥ i for any N ∈ mod op and 1 ≤ i ≤ k− 1, then the socle of the last term in a minimal injective resolution of as a right -module is nonzero.