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Ideal

Definition

A left ideal is a subset $I$ of a ring $R$ which satisfies the following:

• $I$ is a subgroup of the additive group of the ring.
• $\forall a \in I, r \in R: ar \in I$

We denote this as $I \lhd R$.

Right ideals are defined analogously.

Criterion

$\forall i,j \in I; r \in R: i-j \in I; ri \in I$

Properties

In a commutative ring, left- and right-ideals are the same.

The only ideal that is a subring is the ring itself.

Given an ideal $I \lhd R$, we can define the quotient ring $R/I$

Types

Principal ideal: If $I = \{ra: r\in R\}$, or equivalently, $I$ is generated by $a$, we write $I=<a>$ and say $I$ is a principal ideal.

Prime ideal: If $I$ is a proper ideal of $R$, and $a.b \in I \Rightarrow (a\in I$ or $b \in I)$ we say $I$ is a prime ideal, and any generator of $I$ is a prime element of $R$.

Maximal ideal: If $I$ is not strictly contained in any proper ideal of $R$, we say $I$ is a maximal ideal. A maximal ideal is prime.

Examples

In a field, the only ideals are the empty ideal and the entire field (hence there is no such thing [citation needed] as a quotient field)

all these pages adapted with probably insufficient credit from my university's lecture notes