---
tag: 008TBCYDPX
kind: example
name: Zero ring
aliases: [trivial-ring]
msc: [16B99]
instance_of: [0310REJJY8]
uses: [0894J980RK]
sources:
  - url: https://en.wikipedia.org/wiki/Zero_ring
    locator: Definition
same_as:
  wikidata: Q1652969
  wikipedia: Zero ring
---

## Example

The zero ring is the set $\{0\}$ with the operations $0 + 0 = 0$ and $0 \cdot 0 = 0$.
It is a [[commutative ring|0GYGHKX834]] in which $1 = 0$.

## Remarks

A [[ring|0894J980RK]] $R$ is the zero ring if and only if $1 = 0$ in $R$. If $1 = 0$, then $a = a \cdot 1 = a \cdot 0 = 0$ for each $a \in R$.
The zero ring is not an [[integral domain|09S3PFBEPM]] and not a [[field|0QA8CHH8MG]], because these definitions require $1 \neq 0$.

## Explanations

### L1 human

The zero ring has only one element.
All the axioms of a ring hold, but in a trivial way.
It is the only ring in which $1$ and $0$ are the same element.