Rings and modules which are stable under automorphisms of their injective hulls

TL;DR

This paper characterizes rings and modules that remain stable under automorphisms of their injective hulls, establishing new conditions for self-injectivity and pseudo-injectivity, and providing counterexamples and classifications.

Contribution

It proves that prime right nonsingular rings are right self-injective if their modules are automorphism-invariant, and characterizes automorphism-invariant modules as pseudo-injective, answering open questions.

Findings

Prime right nonsingular rings are right self-injective under automorphism-invariance.

Automorphism-invariant modules are exactly pseudo-injective modules.

Counterexample shows the result does not extend to semiprime rings.

Abstract

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.