Ideal

Definition

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

$I$ is a subgroup of the additive group $(R, +)$ .
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.

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Group concept
Ring definition
Ideal definition

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Tag	`0MYSA6MYNA`
URI	`m2://0MYSA6MYNA`
Kind	definition
Writing	W1 stated
MSC	13A15
Defines	Ideal
Uses	Ring, Group

Ideal

Definition

Remarks

Reading path

Where this leads

Further reading

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