Serial ring

Summary

A serial ring is a ring that, as a module over itself on each side, splits into a direct sum of uniserial modules. Serial rings include discrete valuation rings, the rings $\Z/n\Z$, and the rings of upper triangular matrices over a field. The finite-dimensional serial algebras over a field are the Nakayama algebras. Their module theory is very explicit. By a theorem of Warfield, every finitely presented module over a serial ring is a direct sum of uniserial modules.