An ideal is a subset 
of elements in a ring 
that forms an additive
group and has the property that, whenever 
belongs to 
and 
belongs to 
, then 
and 
belong to 
. For example, the set
of even integers is an
ideal in the ring of integers 
. Given an ideal 
, it is possible to define a quotient
ring 
. Ideals are commonly denoted using
a Gothic typeface.
In a commutative ring, a finitely generated ideal is generated by a finite list 
, 
, ..., 
and contains all elements of the form 
, where the coefficients 
are arbitrary elements
of the ring. The list of generators
is not unique, for instance 
in the integers.
In a number ring, ideals can be represented as lattices, and can be given a finite basis of algebraic integers
which generates the ideal additively. Any two bases for
the same lattice are equivalent. A product
ideal is generated additively by the pairwise products of elements
from bases of the two factors. The illustration above shows
an ideal in the Gaussian integers generated by
2 and 
, where elements
of the ideal are indicated in red.
In algebraic geometry, an ideal in a polynomial ring over 
defines an algebraic variety as the common solution
set of its polynomial
equations.
In the full ring of integers of a number field, for every nonzero ideal 
there is a nonzero ideal 
such that
 |
(1)
|
where 
is a nonzero principal ideal, i.e., an ideal generated by one
element. Moreover, there is a finite list of nonzero ideals 
such that this equation may
be satisfied for every nonzero 
. Taking one representative from each inverse ideal class,
the size of this list is known as the class
number. Two nonzero ideals are equivalent when one
is a nonzero scalar multiple
of the other in the number field. The number of these
equivalence classes is the class
number. When the class number is 1, the full
ring of integers has unique
factorization, so the class number measures the
failure of unique factorization in this ring.
Dedekind (1871) showed that every nonzero ideal in the full ring of integers of a number
field is a unique product of nonzero prime ideals. All ideals of 
are principal ideals.
Ideals can be added, multiplied and intersected. The union of ideals usually is not an ideal since it may not be closed
under addition. For ideals in a polynomial
ring over 
,
the addition of ideals corresponds to the intersection
of varieties and the intersection
of ideals corresponds to the union of varieties.
Also, the multiplication of ideals corresponds
to the union of varieties.
Intersection and multiplication are different, for instance consider the ideal 
in ![Z[x,y]](https://mathworld.wolfram.com/images/equations/Ideal/Inline29.svg)
. Then
 |
(2)
|
Sometimes they are the same. If 
,
then
 |
(3)
|
There is also an analog of division, the ideal quotient 
,
and there is an analog of the radical, also called the
ideal radical 
. Given a ring homomorphism 
, ideals in 
extend to ideals in 
, while ideals in 
contract to ideals in
.
The following formulas summarize operations on ideals in commutative rings, where 
denotes ideal contraction, 
denotes ideal extension,
and 
denotes an ideal
quotient:
 |
(4)
|
 |
(5)
|
 |
(6)
|
 |
(7)
|
 |
(8)
|
 |
(9)
|
 |
(10)
|
 |
(11)
|
 |
(12)
|
 |
(13)
|
 |
(14)
|
 |
(15)
|
 |
(16)
|
 |
(17)
|
 |
(18)
|
 |
(19)
|
 |
(20)
|
 |
(21)
|
 |
(22)
|
 |
(23)
|
 |
(24)
|
 |
(25)
|
If 
is an algebra
with an identity element, a left
ideal (respectively, a right ideal) of 
is a subspace 
of 
such that 
whenever 
and 
.
A two-sided ideal is a subset
of 
that is both a left
ideal and a right ideal. For each such algebra 
and an element 
, the sets 
and 
are examples of left
ideals and right ideals respectively, and 
is an example of a two-sided ideal.
Every two-sided ideal of an associative algebra is a Mathieu subspace, but not every Mathieu subspace is an ideal (Zhao 2012).
