---
tag: TW3ESQB3GQ
kind: concept
name: Serial ring
msc: [16P70, 16L30, 16G10]
specializes: [0B1261B5ED]
sources:
  - url: https://en.wikipedia.org/wiki/Serial_module
---

## Summary

A serial ring is a [[ring|0B1261B5ED]] that, as a module over itself on each side, splits into a direct sum of [[uniserial modules|Z1045A0V29]].
Serial rings include [[discrete valuation rings|MGM3CNJW9V]], the rings $\Z/n\Z$, and the rings of [[upper triangular matrices|0N4XY7Z3NN]] over a field.
The finite-dimensional serial algebras over a field are the [[Nakayama algebras|APRFMTZ10K]].
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.

## Explanations

### L1 human

For a serial ring, the ring itself can be cut into a few pieces, and each piece has submodules that line up in a single chain.
So the whole ring is a sum of "ladders" with no branching.
The ring $\Z/12\Z$ is serial: it splits as $\Z/4\Z \times \Z/3\Z$, and each factor has a chain of ideals.
The ring $k[x, y]/(x, y)^2$ over a field $k$ is not serial, because the ideals generated by $x$ and by $y$ cannot be compared.