Definition

For a ring $R$ , a left (right) ideal $I \subset R$ is a subset which is a subgroup with respect to $+$ and satisfies

r i \in I (resp. i r \in I) \forall i \in I and r \in R

A (two-sided) ideal is a subset that is both a left and a right ideal.

NOTE

We do NOT requite that ideals contain $1_{R}$ (unlike subrings) since if $1_{R} \in I$ then $I = R$ .

Generated ideals

Given a ring $R$ and a subset ${x_{i}}_{i \in I}$ , the left ideal generated by ${x_{i}}_{i \in I}$ is the ideal

{i \in I finite \sum r_{i} \cdot x_{i} ∣ r_{i} \in R}

The right ideal generated by a subset can be defined analogously.

The ideal generated by ${x_{i}}_{i \in I}$ (often denoted $(x_{i})_{i \in I}$ ) is the subset

{i \in I finite \sum r_{i} \cdot x_{i} \cdot s_{i} ∣ r_{i}, s_{i} \in R}

For an ideal $I \subset R$ :

$I$ is principal if it is generated by a single element. That is, if $I = (r)$ for some $r \in R$ .

$I$ is finitely generated if $I = (r_{1}, \dots, r_{n})$ for some (finite) $r_{1}, \dots, r_{n} \in R$ .

$I ⊊ R$ is maximal if for any other ideal $J$ such that $I \subseteq J$ , then $J = R$ or $J = I$ .

$I \subset R$ is prime if whenever $ab \in I$ for $a, b \in R$ then either $a \in I$ or $b \in I$ .

An ideal $I \subset R$ is maximal if and only if $R / I$ is a field.

An ideal $I \subset R$ is prime if and only if $R / I$ is an integral domain.