Homological Properties Relative to Injectively Resolving Subcategories

摘要

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