Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3.
An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group.
For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals.
There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory).
Ernst Kummer invented the concept of ideal numbers to serve as the "missing" factors in number rings in which unique factorization fails; here the word "ideal" is in the sense of existing in imagination only, in analogy with "ideal" objects in geometry such as points at infinity.
[1] In 1876, Richard Dedekind replaced Kummer's undefined concept by concrete sets of numbers, sets that he called ideals, in the third edition of Dirichlet's book Vorlesungen über Zahlentheorie, to which Dedekind had added many supplements.
Given a ring R, a left ideal is a subset I of R that is a subgroup of the additive group of
If the ring is commutative, the three definitions are the same, and one talks simply of an ideal.
[6] (It is an instance of a congruence relation and is a generalization of modular arithmetic.)
[8] Conversely, the kernel of a ring homomorphism is a two-sided ideal.
For a rng R, a left ideal I is a subrng with the additional property that
To simplify the description all rings are assumed to be commutative.
The non-commutative case is discussed in detail in the respective articles.
See their respective articles for details: The sum and product of ideals are defined as follows.
right) ideals of a ring R, their sum is which is a left (resp.
The three operations of intersection, sum (or join), and product make the set of ideals of a commutative ring into a quantale.
in the following two cases (at least) (More generally, the difference between a product and an intersection of ideals is measured by the Tor functor:
[18] It can then be shown that every nonzero ideal of a Dedekind domain can be uniquely written as a product of maximal ideals, a generalization of the fundamental theorem of arithmetic.
[19][20][21] Ideals appear naturally in the study of modules, especially in the form of a radical.
By definition, a primitive ideal of R is the annihilator of a (nonzero) simple R-module.
There is also another characterization (the proof is not hard): For a not-necessarily-commutative ring, it is a general fact that
is (see the link) and so this last characterization shows that the radical can be defined both in terms of left and right primitive ideals.
The following simple but important fact (Nakayama's lemma) is built-in to the definition of a Jacobson radical: if M is a module such that
Since a nonzero finitely generated module admits a maximal submodule, in particular, one has: A maximal ideal is a prime ideal and so one has where the intersection on the left is called the nilradical of R. As it turns out,
is also the set of nilpotent elements of R. If R is an Artinian ring, then
need not be an ideal in B (e.g. take f to be the inclusion of the ring of integers Z into the field of rationals Q).
Many classic examples of this stem from algebraic number theory.
then: Remark: Let K be a field extension of L, and let B and A be the rings of integers of K and L, respectively.
of A under extension is one of the central problems of algebraic number theory.
An ideal can also be thought of as a specific type of R-module.