Claim 1: Any injective module is the injective hull of a direct sum of cyclic modules.
Proof: This is an easy Zorn's lemma type argument. Check out Lam's solution to exercise 3.22 in "Exercises in Modules and Rings" for the full argument.
Claim 2: The following are equivalent.
(1) $R$ is a right noetherian ring.
(2) Any direct sum of injective right $R$-modules is injective.
Proof: This is due to Bass and Papp. See Theorem 3.46 in Lam's "Lectures on Modules and Rings".
Thus if $R$ is a right noetherian ring that also satisfies the condition that the injective hull of a cyclic right $R$-module is projective, then we see that every injective right $R$-module is projective (by using claims 1 and 2). This is one of the characterizations of the quasi-Frobenius rings (see this wiki article), so $R$ is noetherian and self-injective on both sides.