Subring

Definition

Let $R$ be a ring. 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.