Rings whose (proper) cyclic modules have cyclic automorphism-invariant hulls
A module \(M\) is called automorphism-invariant if \(M\) is invariant under any automorphism of the injective hull of \(M\). It is shown that for a ring \(R\), if every cyclic right \(R\)-module is automorphism-invariant, then \(R\) is stably-finite (i.e., \(\text{Mat}_n(R)\) is directly-finite for each positive integer \(n\)). The minimal automorphism-invariant essential extension of a module \(M\) is called the automorphism-invariant hull of \(M\). Further, a ring \(R\) is called right \(a\)-hypercyclic if each cyclic right \(R\)-module has a cyclic automorphism-invariant hull. Then the authors show that if a ring \(R\) is right \(a\)-hypercyclic and \(R_R\) has Krull dimension (in the sense of Rentschler and Gabriel), then \(R\) is right Artinian.
scientific article; zbMATH DE number 7377732
scientific article