A module which is invariant under automorphisms of its injective hull is called an automorphism-invariant module. Automorphism-invariant modules, which generalize the notion of quasi-injective modules, are precisely the pseudo-injective modules. We study modules which are invariant under idempotent endomorphisms of their injective hulls and such modules will be called idempotent-invariant modules.
We establish that idempotent-invariant modules, which also generalize the notion of quasi-injective modules, are precisely the quasi-continuous modules and show that the classes of automorphism-invariant modules and idempotent-invariant modules are not contained, one in the other. Some facts and results of this class of modules are obtained. Automorphism-invariant modules are clean but not so for idempotent-invariant modules. We investigate certain conditions under which idempotent-invariant modules can actually be clean.
