  We introduce and study a nontrivial generalization of uniserial modules and
rings. A module is called weakly uniserial if its submodules are comparable
regarding embedding. Also, a right (resp., left) weakly uniserial ring is a
ring which is weakly uniserial as a right (resp., left) module over itself. In
this paper, in addition to providing the properties of weakly uniserial modules
and rings, we show that every right R-module is weakly uniserial if and only if
every 2-generated right R-module is weakly uniserial, if and only if R is a
simple Artinian ring. Then it is determined which torsion-free abelian groups
of rank 1 are weakly uniserial. Finally, when R is a commutative principal
ideal domain, the structure of finitely generated weakly uniserial R-modules
are completely determined.
