Ideal (mathematics)

In ring theory, an ideal is defined as a subset of a ring with the following properties:

1. Zero is an element of the ideal
2. The ideal is closed under addition
3. The product of an element of the ideal and an element of the initial ring is an element of the ideal

In general, one must distinguish between left and right ideals, because many rings do not have commutative multiplication. All discussion here applies to commutative rings.

Examples

An example is the set of even integers, as a subset of the ring of integers. This is an ideal because:

1. 0 is even.
2. The sum of two even integers is even.
3. The product of an even integer with and any other integer is even.

This example is itself an example of a principal ideal. Given any ring R and element <math>x \in R</math>, we may define an ideal (r), which consists of all elements of R which may be written as the product of r with some other element a of the ring. The ideal (r) is called a principal ideal. It is an ideal, because:

1. <math> 0 = 0 \cdot r \in (r)</math>
2. If <math>x \in (r)</math> and <math>y \in (r)</math>, then we can write <math>x = ra</math> and <math>y = rb</math>. Then <math>x+y = ra + rb = r(a+b) \in R</math>.
3. If <math>x \in (r)</math> and <math>b \in R</math> is an arbitrary element of the ring <math>R</math>, then <math>xb = (ar)b = (ab)r \in R</math>.

When <math>R=\mathbb Z</math> is the ring of integers, then every ideal is a principal ideal: that is, any ideal <math>I</math> of the ring of integers is the set of elements "multiples of k" for some integer k. Many familiar rings have the property that every ideal is a principal ideal: such rings are known as principal ideal domains.