Ring theory MOC

Ideal

A subrng is called a left ideal iff , a right ideal iff , 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 . Similarly to a normal subgroup in group theory, an ideal can be used to construct a Quotient ring.

Ideal test

Let be a inhabited subset of a ring . Then is an ideal iff for all and for all and .

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 , a set , or both using angle bracket

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
• Fractional ideal
• Unique factorization of ideals
• Relatively prime ideals

Classification

• Prime ideal
• Maximal ideal
• Principal ideal

Properties

• An ideal is an -module

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