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
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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 . Then
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Sometimes they are the same. If , then
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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:
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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.