---
tag: 249ZVY6WK3
kind: definition
name: Serial ring
msc: [16P70, 16L30, 16G10]
defines: TW3ESQB3GQ
uses: [0B1261B5ED, Z1045A0V29, 0NZ8JA25WK]
sources:
  - url: https://en.wikipedia.org/wiki/Serial_module
---

## Definition

Let $R$ be a [[ring|0B1261B5ED]] with identity.
A right $R$-[[module|0NZ8JA25WK]] is *serial* if it is a direct sum of [[uniserial modules|Z1045A0V29]].
The ring $R$ is *right serial* if the right module $R_R$ is serial.
It is *left serial* if the left module ${}_R R$ is serial.
It is *serial* if it is both right serial and left serial.

## Remarks

Example: let $k$ be a field and $R = k[x]/(x^n)$ with $n \ge 1$.
Its ideals are the ideals $(x^i)/(x^n)$ for $0 \le i \le n$, and they form a chain.
So $R$ is uniserial over itself on both sides, and $R$ is serial.

Non-example: let $R = k[x, y]/(x, y)^2$.
This is a local ring, so $R$ has no idempotents other than $0$ and $1$, and $R$ is indecomposable as a module over itself.
If it were serial, it would therefore be uniserial.
But the ideals $(x)$ and $(y)$ of $R$ are not comparable, so $R$ is not serial.

Equivalently, $R$ is right serial when $R = e_1 R \oplus \cdots \oplus e_m R$ for some orthogonal idempotents $e_1, \dots, e_m$ with each $e_i R$ uniserial, since a direct summand of $R_R$ has the form $eR$ for an idempotent $e$.
A finite-dimensional serial algebra over a field is also called a [[Nakayama algebra|APRFMTZ10K]].