On the Grade of Modules Over Noetherian Rings

ISSN： 0092-7872

Source： Communications in Algebra, Vol.36, Iss.10, 2008-10, pp. : 3616-3631

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Abstract

Let Λ be a left and right noetherian ring and mod Λ the category of finitely generated left Λ-modules. In this article, we show the following results. (1) For a positive integer k, the condition that the subcategory of mod Λ consisting of i-torsionfree modules coincides with the subcategory of mod Λ consisting of i-syzygy modules for any 1 ≤ i ≤ k is left-right symmetric. (2) If Λ is an ∞-Gorenstein ring and N is in mod Λop with grade N = 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 [image omitted] for any M ∈ mod Λ and [image omitted] for any N ∈ mod Λop 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 nonzero.