On automorphism-invariant modules

Abstract

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.

Rings with each right ideal automorphism-invariant

“…By Facts 5.3, 5.4 and 5.5, X is quasi-injective, Y is automorphism-invariant square-free which is orthogonal to X, and X and Y are relatively injective. By [37], U is automorphism-invariant. This shows that each essential right ideal of R is automorphism-invariant.…”

“…Recently, Guil Asensio, Keskin Tütüncü and Srivastava [13] have initiated the study of a more general theory of modules invariant under automorphisms of their covers and envelopes. See [1], [15], [16], [37] and [38] for more details on automorphism-invariant modules.…”