Almost uniserial rings and modules.
Generalizing the notions of uniserial modules and uniserial rings, the authors in this paper, introduce the notions of almost uniserial modules and almost uniserial rings and investigate their properties. They define an \(R\)-module \(M\) to be almost uniserial if any two non-isomorphic submodules of \(M\) are comparable and a ring \(R\) is said to be almost left (resp. right) uniserial if \(R\), as a left (resp. right) \(R\)-module, is almost uniserial. They prove that an almost left uniserial ring \(R\) is either local or every maximal left ideal of \(R\) is cyclic. From this they deduce that every Noetherian, almost left uniserial ring is either a local ring or a principal left ideal ring. They also prove that a left Artinian, principal left ideal ring \(R\) is almost left uniserial if and only if either \(R\) is left uniserial or \(R\) is isomorphic to the matrix ring \(R=M_2(D)\) for some division ring \(D\). The authors conclude the paper with a structure theorem for commutative Artinian almost uniserial rings.
scientific article; zbMATH DE number 6509815
scientific article