Published online by Cambridge University Press: 20 November 2018
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.