Injective envelopes and flat covers of Matlis reflexive modules
We show that for a commutative noetherian local ring R, every Matlis reflexive R-module has a reflexive injective envelope if and only if every Matlis reflexive R-module has a reflexive flat cover. This occurs if and only if R is complete and has Krull dimension less than or equal to 1. We also exhibit a family of Matlis reflexive R-modules whose injective envelopes are not Matlis reflexive.
Belshoff, Richard G., and Jinzhong Xu. "Injective envelopes and flat covers of Matlis reflexive modules." Journal of pure and applied algebra 79, no. 3 (1992): 205-215.
Journal of Pure and Applied Algebra