A note on the Matlis dual of a certain injective hull

Let (R,m)(R,m) denote a local ring with E=ER(R/m)E=ER(R/m) the injective hull of the residue field. Let p∈SpecR denote a prime ideal with dim⁡R/p=1dim⁡R/p=1, and let ER(R/p)ER(R/p) be the injective hull of R/pR/p. As the main result we prove that the Matlis dual HomR(ER(R/p),E)HomR(ER(R/p),E) is isomorphic to Rpˆ, the completion of RpRp, if and only if R/pR/p is complete. In the case of R   a one dimensional domain there is a complete description of Q⊗RRˆ in terms of the completion Rˆ.