---
tag: APRFMTZ10K
kind: concept
name: Nakayama algebra
msc: [16G20, 16G60]
specializes: [3CTW1P4HBP]
sources:
  - url: https://en.wikipedia.org/wiki/Nakayama_algebra
same_as:
  wikidata: Q6960389
  wikipedia: Nakayama algebra
---

## Summary

A Nakayama algebra is a finite-dimensional algebra whose indecomposable projective modules, on both sides, are [[uniserial|Z1045A0V29]]: their submodules form a single chain.
Nakayama studied them under the name generalized uniserial algebras, and they are now named after him.
Every indecomposable module over a Nakayama algebra is uniserial, so the algebra has only finitely many indecomposable modules up to isomorphism.
They are one of the few classes of algebras whose modules can be listed completely, and they serve as a test case in representation theory.

## Explanations

### L1 human

A module is uniserial when its submodules line up in one chain, like the rungs of a ladder.
A Nakayama algebra is an algebra whose basic projective building blocks all have this ladder shape.
The algebra $K[x]/(x^n)$ is the simplest example: its ideals are $(x^0) \supset (x^1) \supset \dots \supset (x^n) = 0$, one chain.
The path algebra of a quiver that is a single directed line $1 \to 2 \to \dots \to n$ is another example.
Because everything is a ladder, every indecomposable module is determined by its top layer and its length.