Homological Properties Relative to Injectively Resolving Subcategories

摘要

Let $mathcal{E}$ be an injectively resolving subcategory ofleft $R$-modules. A left $R$-module $M$(resp. right $R$-module $N$) is called $mathcal{E}$-injective(resp. $mathcal{E}$-flat)if $operatorname{Ext}_R^1(G,M)=0$ (resp. $operatorname{Tor}_1^R(N,G)=0$)for any $Ginmathcal{E}$.Let $mathcal{E}$ be a covering subcategory.We prove that a left $R$-module $M$ is $mathcal{E}$-injectiveif and only if $M$ is a direct sumof an injective left $R$-module and a reduced $mathcal{E}$-injectiveleft $R$-module.Suppose $mathcal{F}$ is a preenveloping subcategory of right$R$-modules such that$mathcal{E}^+subseteqmathcal{F}$ and $mathcal{F}^+subseteqmathcal{E}$.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 $mathcal{E}$-flatmodules.