Weakly (uni)serial modules

In this paper, we continue to study weakly uniserial modules and rings. A module is called weakly uniserial if its submodules are comparable regarding embedding. Also, a right weakly uniserial ring is a ring which is weakly uniserial as a right module over itself. In addition to providing more properties of these modules, we introduce and investigate weakly serial modules (which are a direct sum of weakly uniserial modules). A right weakly serial ring is a ring which is weakly serial as a right module over itself. It is shown that over a right Artinian local ring [Formula: see text], every (2-generated) right [Formula: see text]-module with a semiprime annihilator is weakly uniserial. The converse is true when [Formula: see text] is a commutative Noetherian ring. We prove that an injective [Formula: see text]-module is weakly uniserial if and only if it is isomorphic to [Formula: see text], for some prime number [Formula: see text]. In a weakly serial module we show that every its nonzero submodule contains a weakly uniserial submodule and every its fully invariant submodule is weakly serial. Also, it is shown that a right duo right self-injective ring is right weakly serial if and only if every projective right module is (weakly) serial.