---
tag: 0859D58WPD
kind: definition
name: Subring
msc: [16B99]
defines: 0GNCRPYXYV
uses: [0894J980RK]
sources:
  - url: https://en.wikipedia.org/wiki/Subring
    locator: Definition
---

## Definition

Let $R$ be a [[ring|0894J980RK]].
A subring of $R$ is a subset $S \subseteq R$ with these properties.

1. $1 \in S$.
2. If $a, b \in S$, then $a - b \in S$.
3. If $a, b \in S$, then $a \cdot b \in S$.

A subring $S$ is a ring with the addition and the multiplication of $R$, restricted to $S$.

## Remarks

The integers $\mathbb{Z}$ are a subring of $\mathbb{Q}$, and $\mathbb{Q}$ is a subring of $\mathbb{R}$.
The set $2\mathbb{Z}$ of even integers is closed under addition and multiplication, but it is not a subring of $\mathbb{Z}$, because $1 \notin 2\mathbb{Z}$.
Authors who do not require a ring to have an identity also do not require $1 \in S$.

## Explanations

### L1 human

A subring is a smaller ring inside a ring, with the same operations.
For example, the integers sit inside the rational numbers.
The sum, the difference and the product of two integers are again integers.