Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by an integer results in an even number; these closure and absorption properties are the defining properties of an ideal. 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.
Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. 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).
The related, but distinct, concept of an ideal in order theory is derived from the notion of ideal in ring theory. A fractional ideal is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity.
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. 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. Later the notion was extended beyond number rings to the setting of polynomial rings and other commutative rings by David Hilbert and especially Emmy Noether.
A right ideal is defined with the condition "r x ∈ I" replaced by "x r ∈ I". A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. In the language of modules, the definitions mean that a left (resp. right, two-sided) ideal of R is precisely a left (resp. right, bi-) R-submodule of R when R is viewed as an R-module. When R is a commutative ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.
To understand the concept of an ideal, consider how ideals arise in the construction of rings of "elements modulo". For concreteness, let us look at the ring ℤn of integers modulo a given integer n ∈ ℤ (note that ℤ is a commutative ring). The key observation here is that we obtain ℤn by taking the integer line ℤ and wrapping it around itself so that various integers get identified. In doing so, we must satisfy 2 requirements:
The second requirement forces us to make additional identifications (i.e., it determines the precise way in which we must wrap ℤ around itself). The notion of an ideal arises when we ask the question:
What is the exact set of integers that we are forced to identify with 0?
Remark. Identifications with elements other than 0 also need to be made. For example, the elements in 1 + nℤ must be identified with 1, the elements in 2 + nℤ must be identified with 2, and so on. Those, however, are uniquely determined by nℤ since ℤ is an additive group.
Therefore, an ideal I of a commutative ring R captures canonically the information needed to obtain the ring of elements of R modulo a given subset S ⊆ R. The elements of I, by definition, are those that are congruent to zero, that is, identified with zero in the resulting ring. The resulting ring is called the quotient of R by I and is denoted R/I. Intuitively, the definition of an ideal postulates two natural conditions necessary for I to contain all elements designated as "zeros" by R/I:
It turns out that the above conditions are also sufficient for I to contain all the necessary "zeros": no other elements have to be designated as "zero" in order to form R/I. (In fact, no other elements should be designated as "zero" if we want to make the fewest identifications.)
Remark. The above construction still works using two-sided ideals even if R is not necessarily commutative.
(For the sake of brevity, some results are stated only for left ideals but are usually also true for right ideals with appropriate notation changes.)
To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles.
Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings. Different types of ideals are studied because they can be used to construct different types of factor rings.
Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details:
If a product is replaced by an intersection, a partial distributive law holds:
Remark: The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given ring forms a complete modular lattice. The lattice is not, in general, a distributive lattice. The three operations of intersection, sum (or join), and product make the set of ideals of a commutative ring into a quantale.
In the first computation, we see the general pattern for taking the sum of two finitely generated ideals, it is the ideal generated by the union of their generators. In the last three we observe that products and intersections agree whenever the two ideals intersect in the zero ideal. These computations can be checked using Macaulay2.
Ideals appear naturally in the study of modules, especially in the form of a radical.For simplicity, we work with commutative rings but, with some changes, the results are also true for non-commutative rings.