---
tag: KRTDAZP29B
kind: definition
name: Noncommutative ring
msc: [16B99]
defines: D1SA8CHJV0
uses: [0894J980RK]
sources:
  - url: https://en.wikipedia.org/wiki/Noncommutative_ring
---

## Definition

Let $R$ be a [[ring|0894J980RK]].
The ring $R$ is *noncommutative* if there are elements $a, b \in R$ with $ab \ne ba$.

## Remarks

Example: the ring $M_2(\R)$ of $2 \times 2$ real matrices is noncommutative.
With $E_{12}$ the matrix with $1$ in position $(1,2)$ and $0$ elsewhere, and $E_{21}$ defined likewise, one has $E_{12} E_{21} = E_{11} \ne E_{22} = E_{21} E_{12}$.
Non-example: the ring $\Z$ is not noncommutative, since $ab = ba$ for all integers $a, b$.
Many authors use "noncommutative ring" for a ring that is not assumed to be commutative.
In that use, commutative rings are allowed as special cases.