---
tag: AWSBZWWMG9
kind: concept
name: Endomorphism ring
msc: [16S50]
sources:
  - url: https://en.wikipedia.org/wiki/Endomorphism_ring
same_as:
  wikidata: Q2896709
  wikipedia: Endomorphism ring
---

## Summary

The endomorphism ring of a module is the set of all module homomorphisms from the module to itself, with pointwise addition and composition as multiplication.
It is the main source of noncommutative rings: every ring is a subring of the endomorphism ring of its own additive group.
Properties of a module are often read off its endomorphism ring.
For example, a nonzero module is indecomposable exactly when its endomorphism ring has no idempotents other than $0$ and $1$.
For a vector space it is the [[algebra of endomorphisms|0E3X2G1ZQB]], and for $K^n$ it is the [[matrix ring|0FHVPRAE60]] $M_n(K)$.

## Explanations

### L1 human

The structure-preserving maps from an object to itself can be added and composed.
Composition plays the role of multiplication, and the identity map plays the role of $1$.
For the integers as an abelian group, every such map is multiplication by a fixed integer, so the endomorphism ring is $\Z$ again.
For the plane $\R^2$ as a real vector space, the endomorphisms are the linear maps, and they form the ring of $2 \times 2$ real matrices.