---
tag: 0GYGHKX834
kind: definition
name: Commutative ring
msc: [13A99]
defines: 0310REJJY8
uses: [0894J980RK]
sources:
  - url: https://en.wikipedia.org/wiki/Commutative_ring
    locator: Definition
---

## Definition

A commutative ring is a [[ring|0894J980RK]] $R$ such that $a \cdot b = b \cdot a$ for all $a, b \in R$.

## Remarks

The integers $\mathbb{Z}$, the rational numbers $\mathbb{Q}$ and the polynomials in one variable over $\mathbb{Q}$ are commutative rings.
The ring of $2 \times 2$ real matrices is not commutative.
Every [[field|0QA8CHH8MG]] is a commutative ring.

## Explanations

### L1 human

In a commutative ring, the order of the factors in a product does not matter.
The integers are the model: $3 \cdot 5 = 5 \cdot 3$.