---
tag: 0BXA4Q0JHW
kind: statement
role: theorem
name: Ring of integers
msc: [13A99]
uses: [0C4JS9RF5M]
sources:
  - url: https://en.wikipedia.org/wiki/Integer
    locator: Algebraic properties
---

## Statement

The [[integers|0C4JS9RF5M]] $\mathbb{Z}$, with their addition $+$ and multiplication $\cdot$, form a commutative ring with identity $1$.
That is, for all integers $a$, $b$ and $c$:

1. $(a + b) + c = a + (b + c)$ and $a + b = b + a$;
2. $a + 0 = a$, and there is an integer $-a$ with $a + (-a) = 0$;
3. $(a b) c = a (b c)$ and $a b = b a$;
4. $a \cdot 1 = a$;
5. $a (b + c) = a b + a c$.

## Remarks

The ring $\mathbb{Z}$ is not a field: for example, $2$ has no multiplicative inverse in $\mathbb{Z}$.
The ring $\mathbb{Z}$ has no zero divisors: if $a b = 0$, then $a = 0$ or $b = 0$. Thus it is an integral domain.
Here "ring of integers" means the ring $\mathbb{Z}$. In algebraic number theory, the same words have a more general meaning.

## Explanations

### L1 human

This statement lists the usual rules of computation with whole numbers.
You can add, subtract and multiply integers, and the result is always an integer.
Division is different: $1 / 2$ is not an integer.