---
tag: 0JHM6JK7FD
kind: example
name: The ring of integers
msc: [13A99]
instance_of: [0310REJJY8]
uses: [0GYGHKX834]
sources:
  - url: https://en.wikipedia.org/wiki/Integer
    locator: Algebraic properties
---

## Example

Let $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$ be the set of integers, with the usual addition and multiplication.
Then $\mathbb{Z}$ is a [[commutative ring|0GYGHKX834]].
Its zero element is $0$ and its identity element is $1$.

## Remarks

The only [[units|019NFGFPR5]] of $\mathbb{Z}$ are $1$ and $-1$.
The product of two nonzero integers is not zero, thus $\mathbb{Z}$ is an [[integral domain|09S3PFBEPM]].
The ring $\mathbb{Z}$ is not a [[field|0QA8CHH8MG]], because $2$ has no multiplicative inverse in $\mathbb{Z}$.

## Explanations

### L1 human

The integers are the first example of a ring.
You can add, subtract and multiply two integers, and the result is an integer.
You cannot always divide: $1 / 2$ is not an integer.