On automorphism-invariant modules

Let M and N be two modules. M is called automorphism N-invariant if for any essential submodule A of N, any essential monomorphism f : A → M can be extended to some g ∈ Hom(N, M). M is called automorphism-invariant if M is automorphism M-invariant. This notion is motivated by automorphism-invariant modules analog discussed in a recent paper by Lee and Zhou [Modules which are invariant under automorphisms of their injective hulls, J. Algebra Appl. 12(2) (2013), 1250159, 9 pp.]. Basic properties of mutually automorphism-invariant modules and automorphism-invariant modules are proved and their connections with pseudo-injective modules are addressed.