[[Ring theory MOC]]
# Ideal

A [[subrng]] $A \leq R$ is called a **left ideal** iff $RA \sube A$,
a **right ideal** iff $AR \sube A$,
and a **two-sided ideal** (sometimes just **ideal**) iff both conditions hold. #m/def/ring 
This property is sometimes called absorption,
and is equivalent to being a (left/right/two-sided) [[submodule]] of $R$.
Similarly to a [[normal subgroup]] in group theory,
an ideal can be used to construct a [[Quotient ring]].

> [!tip]+ Ideal test
> Let $A$ be a inhabited subset of a ring $R$.
> Then $A$ is an ideal iff $a-b \in A$ for all $a,b \in A$
> and $ar, ra \in A$ for all $a \in A$ and $r \in R$. 

See [[algebra ideal]] for the similar concept for an algebra over a field.
Ideals began with Albert Kummer's [[Ideal number]], which Dedekind realized could be captured using the ideal-as-set formulation.

In a number theoretic context, it is usual to denote the ideal generated by an element $x \in R$, a set $A \sube R$, or both using angle bracket
$$
\begin{align*}
\langle A, x \rangle = \langle A, x \rangle _{\trianglelefteq R}
\end{align*}
$$

## Ideal arithmetic

When working with an [[integral domain]] it useful to generalize to a [[fractional ideal]],
whence ideals are referred to as **integral ideals**.

- [[Product ideal]] $\langle\mathfrak{a}\mathfrak{b}\rangle$
- [[Fractional ideal]]
- [[Unique factorization of ideals]]
- [[Relatively prime ideals]]

## Classification

- [[Prime ideal]]
- [[Maximal ideal]]
- [[Principal ideal]]

## Properties

- An ideal $I \trianglelefteq R$ is an $R$-[[module]]

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