Abstract:
A uniserial module is a module whose submodule lattice is a chain. A serial module is a direct sum of uniserial modules. A ring is said to be serial if it is a serial module both as a left and right module over itself. Nakayama showed that if a ring is Artinian serial, then all left modules are serial. Later, the converse of the above result that if all modules over a ring are serial, then the ring is serial Artinian was proved by Skornyakov. It is then natural to ask about the existence of rings such that all modules over it can be embedded into serial modules. Tuganbaev answered this question in the semi-primary case: All left (resp. right) modules over a semi-primary ring embed into serial left (resp. right) modules if and only if the ring is serial Noetherian. In this talk, we look at other cases such as finite dimensional algebras, perfect rings, commutative (local) Noetherian rings, and commutative von Neuman regular rings.