Almost uniserial rings and modules

We study the class of almost uniserial rings as a straightforward common generalization of left uniserial rings and left principal ideal domains. A ring R is called almost left uniserial if any two non-isomorphic left ideals of R are linearly ordered by inclusion, i.e., for every pair I, J of left ideals of R either I⊆J, or J⊆I, or I≅J. Also, an R-module M is called almost uniserial if any two non-isomorphic submodules are linearly ordered by inclusion. We give some interesting and useful properties of almost uniserial rings and modules. It is shown that a left almost uniserial ring is either a local ring or its maximal left ideals are cyclic. A Noetherian left almost uniserial ring is a local ring or a principal left ideal ring. Also, a left Artinian principal left ideal ring R is almost left uniserial if and only if R is left uniserial or R=M2(D), where D is a division ring. Finally we consider Artinian commutative rings which are almost uniserial and we obtain a structure theorem for these rings.