---
tag: 0894J980RK
kind: definition
name: Ring
msc: [16B99]
defines: 0B1261B5ED
uses: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Ring_(mathematics)
    locator: Definition
---

## Definition

A ring is a set $R$ with two binary operations and two elements $0, 1 \in R$ that satisfy the axioms below.
The operations are the addition $+$ and the multiplication $\cdot$.

1. $(R, +)$ is an abelian [[group|G5S1M9]] with identity element $0$.
2. The multiplication is associative: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c \in R$.
3. The element $1$ is a multiplicative identity: $1 \cdot a = a \cdot 1 = a$ for all $a \in R$.
4. The multiplication is distributive over the addition: $a \cdot (b + c) = a \cdot b + a \cdot c$ and $(b + c) \cdot a = b \cdot a + c \cdot a$ for all $a, b, c \in R$.

## Remarks

Some authors do not require the element $1$. They call the structure of this definition a ring with identity, or a unital ring.
The axioms give $0 \cdot a = a \cdot 0 = 0$ and $(-a) \cdot b = -(a \cdot b)$ for all $a, b \in R$.
The integers $\mathbb{Z}$, the integers modulo $n$ and the $n \times n$ real matrices are rings.
A ring in which $a \cdot b = b \cdot a$ for all $a, b$ is a [[commutative ring|0GYGHKX834]].

## Explanations

### L1 human

A ring is a place where you can add, subtract and multiply, as with the integers.
You cannot always divide.
The order of a product can matter: for two matrices $A$ and $B$, $AB$ and $BA$ are often different.