On the grade of modules over Noetherian rings

Abstract

Let $Λ$ be a left and right noetherian ring and $\mod Λ$ 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 $\mod Λ$ consisting of $i$-torsionfree modules coincides with the subcategory of $\mod Λ$ consisting of $i$-syzygy modules for any $1\leq i \leq k$ is left-right symmetric. (2) If $Λ$ is an Auslander ring and $N$ is in $\mod Λ^{op}$ with $\grade N=k<\infty$, 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 $\grade {\rm Ext}_Λ^k(M, Λ)\geq k$ for any $M\in\mod Λ$ and $\grade {\rm Ext}_Λ^i(N, Λ)\geq i$ for any $N\in\mod Λ^{op}$ and $1\leq i \leq k-1$, then the socle of the last term in a minimal injective resolution of $Λ$ as a right $Λ$-module is non-zero.