Injective envelopes and flat covers of modules over a commutative ring

AbstractLet R be a commutative semilocal ring and U be the minimal injective cogenerator in the category of R-modules. If R is almost noetherian, i.e., each non-minimal prime ideal of R is finitely generated, we prove that the following three statements are equivalent: (1) every U-reflexive R-module has a U-reflexive injective envelope; (2) every U-reflexive R-module has a U-reflexive flat cover; and (3) R is linearly compact and the Krull dimension of R is less than or equal to 1. Our examples show that this result is a non-trivial generalization of a recent theorem of Belshoff and Xu, and that the condition R being almost noetherian is essential