Ideal

A subrng $𝐴 \leq 𝑅$ is called a left ideal iff $𝑅 𝐴 \subseteq 𝐴$ , a right ideal iff $𝐴 𝑅 \subseteq 𝐴$ , 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.

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 $𝑎 𝑟, 𝑟 𝑎 \in 𝐴$ for all $𝑎 \in 𝐴$ and $𝑟 \in 𝑅$ .

See algebra ideal for the similar concept for an algebra over a ring. 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 $𝐴 \subseteq 𝑅$ , or both using angle bracket

⟨ 𝐴, 𝑥 ⟩ = ⟨ 𝐴, 𝑥 ⟩ ⊴ 𝑅

Ideal arithmetic

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

Classification

Properties

An ideal $𝐼 ⊴ 𝑅$ is an $𝑅$ -module

#state/tidy | #lang/en | #SemBr