Duality of Preenvelopes and Pure Injective Modules

Let $R$ be an arbitrary ring and (-)^+=\Hom_{\mathbb{Z}}(-, \mathbb{Q}/\mathbb{Z}) where $\mathbb{Z}$ is the ring of integers and $\mathbb{Q}$ is the ring of rational numbers, and let $\mathcal{C}$ be a subcategory of left $R$-modules and $\mathcal{D}$ a subcategory of right $R$-modules such that $X^+\in \mathcal{D}$ for any $X\in \mathcal{C}$ and all modules in $\mathcal{C}$ are pure injective. Then a homomorphism $f: A\to C$ of left $R$-modules with $C\in \mathcal{C}$ is a $\mathcal{C}$-(pre)envelope of $A$ provided $f^+: C^+\to A^+$ is a $\mathcal{D}$-(pre)cover of $A^+$. Some applications of this result are given.Comment: 9 pages, to appear in Canadian Mathematical Bulleti

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.Comment: 17 pages. To appear in Communications in Algebr

Proper Resolutions and Gorenstein Categories

Let $\mathscr{A}$ be an abelian category and $\mathscr{C}$ an additive full subcategory of $\mathscr{A}$. We provide a method to construct a proper $\mathscr{C}$-resolution (resp. coproper $\mathscr{C}$-coresolution) of one term in a short exact sequence in $\mathscr{A}$ from that of the other two terms. By using these constructions, we answer affirmatively an open question on the stability of the Gorenstein category $\mathcal{G}(\mathscr{C})$ posed by Sather-Wagstaff, Sharif and White; and also prove that $\mathcal{G}(\mathscr{C})$ is closed under direct summands. In addition, we obtain some criteria for computing the $\mathscr{C}$-dimension and the $\mathcal{G}(\mathscr{C)}$-dimension of an object in $\mathscr{A}$.Comment: 35 pages. arXiv admin note: substantial text overlap with arXiv:1012.170