Ideal (ring theory)

In ring theory, a type of mathematics, an ideal is a subset of a ring that generalizes some properties of sets defined by divisibility, like the even numbers.

Definition

If [math]\displaystyle{ (R,+,\cdot) }[/math] is a ring, then [math]\displaystyle{ I\subseteq R }[/math] is an ideal in [math]\displaystyle{ R }[/math] if it is absorptive under the multiplication operator [math]\displaystyle{ \cdot }[/math]. This means that multiplying any element of the ideal by any element of the ring gives an element of the ideal.

On non-commutative rings, ideals can be left ideals or right ideals. This depends on the order of multiplication, with left ideals being left absorptive [math]\displaystyle{ x\in R,y\in I\implies x\cdot y\in I }[/math] and right ideals being right absorptive [math]\displaystyle{ x\in R,y\in I\implies y\cdot x\in I }[/math] An ideal which is both a left ideal and a right ideal is a two-sided ideal. All ideals of commutative rings are two-sided.

Special cases

Every ring has two trivial two-sided ideals, the zero ideal containing only the additive identity, and the unit ideal, another name for the ring itself. Other ideals are called proper ideals. A ring with no proper two-sided ideals is called a simple ring.

A maximal ideal is an ideal that is not a subset of any other ideals except the unit ideal. A minimal ideal is an ideal that has no ideals as subsets except the zero ideal.

A prime ideal is an ideal that has the additional property

[math]\displaystyle{ ab\in I \implies a\in I \mbox{ or } b\in I }[/math]

In words, this says that if the product of two elements is in the ideal, then at least one of the elements must be in the ideal.

A principal ideal is an ideal containing exactly all the multiples of a single element [math]\displaystyle{ a }[/math]:

[math]\displaystyle{ I = \left\{ax : x\in R\right\} }[/math]