Ideal

Definition

Let $R$ be a ring. A left ideal of $R$ is a subset $I \subseteq R$ with these properties.

1. $I$ is a subgroup of the additive group $(R, +)$.
2. For all $r \in R$ and $a \in I$, $r a \in I$.

If $R$ is a commutative ring, a left ideal is simply called an ideal.

Remarks

In $\mathbb{Z}$, the set $n\mathbb{Z}$ of the multiples of $n$ is an ideal for every integer $n$. Every ring $R$ has the ideals $\{0\}$ and $R$. An ideal contains $1$ if and only if it is all of $R$. In a noncommutative ring, one also uses right ideals and two-sided ideals.

Explanations

L1 human

An ideal is a set that absorbs multiplication. The even integers are an example: an even number times any integer is even. Ideals play the role in rings that normal subgroups play in groups.