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  is a ring, then  is an ideal in  if it is absorptive under the multiplication operator . 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

and right ideals being right absorptive

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

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 :

Category:Ring theory