Ring theory MOC

Ideal

A subrng 𝐴 ≤𝑅 is called a left ideal iff 𝑅𝐴 ⊆𝐴, a right ideal iff 𝐴𝑅 ⊆𝐴, and a two-sided ideal (sometimes just ideal) iff both conditions hold. 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 ∈𝐴 for all 𝑎,𝑏 ∈𝐴 and 𝑎𝑟,𝑟𝑎 ∈𝐴 for all 𝑎 ∈𝐴 and 𝑟 ∈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 ∈𝑅, a set 𝐴 ⊆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.

• Product ideal ⟨𝔞𝔟⟩
• Fractional ideal
• Unique factorization of ideals
• Relatively prime ideals

Classification

• Prime ideal
• Maximal ideal
• Principal ideal

Properties

• An ideal 𝐼 ⊴𝑅 is an 𝑅-module

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