Injective envelopes and flat covers of modules over a commutative ring (Q1916410)

The author proves the following principal result: Let \(R\) be a commutative semi-local ring with Krull dimension. If \(R\) is an almost noetherian ring, i.e., each non-minimal prime ideal of \(R\) is finitely generated and \(U\) is the minimal injective cogenerator in the category of \(R\)-modules, then the following 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. (3) \(R\) is linear compact and \(\dim (R)\leq 1\).