Automorphisms of submodules and their extensions. (Q2852145)

A module \(M\) is called an automorphism-invariant module if \(M\) is invariant under any automorphism of its injective envelope \(E(M)\). Clearly, any quasi-injective module is automorphism-invariant but there exist examples of automorphism-invariant modules that are not quasi-injective.NEWLINENEWLINE It has been shown by the reviewer and Guil Asensio that an automorphism-invariant module \(M\) is quasi-injective if \(\text{End}(M)\) has no homomorphic image isomorphic to the field of two elements. It has also been recently shown by the reviewer, Er and Singh that a module \(M\) is automorphism-invariant if and only if every monomorphism from a submodule \(N\) of \(M\) to \(M\) extends to an endomorphism of \(M\). The reviewer and Guil Asensio have shown that automorphism-invariant modules satisfy the exchange property and are examples of clean modules.NEWLINENEWLINE The paper under review studies a generalization of the notion of automorphism-invariant modules. The author calls a module \(M\) to be automorphism-extendable if for each submodule \(N\) of \(M\), any automorphism of \(N\) extends to an endomorphism of \(M\). It is not too difficult to see that any automorphism-invariant module is automorphism-extendable, however the converse is not true, in general. It is shown that for an Artinian module, the notions of automorphism-extendable and automorphism-invariant coincide. The author also shows that if \(R\) is an Artinian serial ring, then any automorphism-extendable right \(R\)-module is quasi-injective.