  It is proved, among other results, that a prime right nonsingular ring (in
particular, a simple ring) $R$ is right self-injective if $R_R$ is invariant
under automorphisms of its injective hull. This answers two questions raised by
Singh and Srivastava, and Clark and Huynh. An example is given to show that
this conclusion no longer holds when prime ring is replaced by semiprime ring
in the above assumption. Also shown is that automorphism-invariant modules are
precisely pseudo-injective modules, answering a recent question of Lee and
Zhou. Furthermore, rings whose cyclic modules are automorphism-invariant are
investigated.
