Homological Properties Relative to Injectively Resolving Subcategories

Gao, Zenghui

Canadian mathematical bulletin, Tome 58 (2015) no. 4, pp. 741-756

Résumé

Let $\mathcal{E}$ be an injectively resolving subcategory of left $R$ -modules. A left $R$ -module $M$ (resp. right $R$ -module $N$ ) is called $\mathcal{E}$ -injective (resp. $\mathcal{E}$ -flat) if $\text{Ext}_{R}^{1}\left( G,\,M \right)\,=\,0$ (resp. $\text{Tor}_{1}^{R}\left( N,\,G \right)\,=\,0$ ) for any $G\,\in \,\mathcal{E}$ . Let $\mathcal{E}$ be a covering subcategory. We prove that a left $R$ -module $M$ is $\mathcal{E}$ -injective if and only if $M$ is a direct sum of an injective left $R$ -module and a reduced $\mathcal{E}$ -injective left $R$ -module. Suppose $\mathcal{F}$ is a preenveloping subcategory of right $R$ -modules such that ${{\mathcal{E}}^{+}}\,\subseteq \,\mathcal{F}$ and ${{\mathcal{F}}^{+}}\,\subseteq \,\mathcal{E}$ . It is shown that a finitely presented right $R$ -module $M$ is $\mathcal{E}$ -flat if and only if $M$ is a cokernel of an $\mathcal{F}$ -preenvelope of a right $R$ -module. In addition, we introduce and investigate the $\mathcal{E}$ -injective and $\mathcal{E}$ -flat dimensions of modules and rings. We also introduce $\mathcal{E}$ -(semi)hereditary rings and $\mathcal{E}$ -von Neumann regular rings and characterize them in terms of $\mathcal{E}$ -injective and $\mathcal{E}$ -flat modules.

Détail

DOI : 10.4153/CMB-2015-058-3

Mots-clés : 16E30, 16E10, 16E60, injectively resolving subcategory, ε-injective module (dimension), ε-flat module (dimension), cover, preenvelope, ε-(semi)hereditary ring