Homological Properties Relative to Injectively Resolving Subcategories

Abstract

Abstract. Let E be an injectively resolving subcategory of le R-modules.Let E be a covering subcategory. We prove that a le R-module M is E-injective if and only if M is a direct sum of an injective le R-module and a reduced E-injective le R-module. Suppose F is a preenveloping subcategory of right R-modules such that E + ⊆ F and F + ⊆ E. It is shown that a nitely presented right R-module M is E-at 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-at 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-at modules.