On the grade of modules over Noetherian rings

Let $\Lambda$ be a left and right noetherian ring and $\mod \Lambda$ the category of finitely generated left $\Lambda$-modules. In this paper we show the following results: (1) For a positive integer $k$, the condition that the subcategory of $\mod \Lambda$ consisting of $i$-torsionfree modules coincides with the subcategory of $\mod \Lambda$ consisting of $i$-syzygy modules for any $1\leq i \leq k$ is left-right symmetric. (2) If $\Lambda$ is an Auslander ring and $N$ is in $\mod \Lambda ^{op}$ 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 $\Lambda$ as a right $\Lambda$-module. (3) Assume that both the left and right self-injective dimensions of $\Lambda$ are $k$. If $\grade {\rm Ext}_{\Lambda}^k(M, \Lambda)\geq k$ for any $M\in\mod \Lambda$ and $\grade {\rm Ext}_{\Lambda}^i(N, \Lambda)\geq i$ for any $N\in\mod \Lambda ^{op}$ and $1\leq i \leq k-1$, then the socle of the last term in a minimal injective resolution of $\Lambda$ as a right $\Lambda$-module is non-zero.