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.
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. Ideals have multiplication, and
this is basically the Kronecker product of the
two bases. The illustration above shows an ideal in the Gaussian
integers generated by 2 and 
,
where elements of the ideal are indicated in red.
From the perspective of algebraic geometry, ideals correspond to varieties.
For any ideal 
,
there is an ideal 
such that
 |
(1)
|
where 
is a principal
ideal, (i.e., an ideal of rank 1). Moreover there is a finite list of ideals
such that this equation may be satisfied
for every 
. The size of this list is known as the
class number. In effect, the above relation imposes
an equivalence relation on ideals, and the
number of ideals modulo this relation is the class number.
When the class number is 1, the corresponding number
ring has unique factorization and, in a sense, the class
number is a measure of the failure of unique factorization in the original number
ring.
In 1871, Dedekind showed that every nonzero ideal in the domain of integers of a field
is a unique product of prime ideals, and in fact all
ideals of 
are of this form and therefore 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. From the perspective of algebraic geometry, 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]](/images/equations/Ideal/Inline27.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, where 
denotes contract, 
denotes ideal
extension, and 
denotes an ideal quotient:
 |
(4)
|
 |
(5)
|
 |
(6)
|
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(7)
|
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(8)
|
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(9)
|
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(10)
|
 |
(11)
|
 |
(12)
|
 |
(13)
|
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(14)
|
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(15)
|
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(16)
|
 |
(17)
|
 |
(18)
|
 |
(19)
|
 |
(20)
|
 |
(21)
|
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(22)
|
 |
(23)
|
 |
(24)
|
 |
(25)
|
If 
is an algebra, a left (right) ideal
of 
is a subspace 
of 
such that 
whenever 
and 
.
A two-sided ideal is a subset of 
that is both a left and right ideal. For each algebra 
and an element 
, the sets 
and 
are examples of left and right ideals respectively,
and 
is an example of a
two-sided ideal.
