Ideal (ring theory)

In abstract algebra, an ideal is a subset I of a ring R which is closed under the ring operations in the following sense:

for any a, b in I, we have a + b in I;
for any a in I and r in R, we have ra in I and ar in I.

If the ring is not commutative, these ideals are sometimes called two-sided to distinguish them from the left-sided ideals (where only ra in I is required in the second condition) and the similarly defined right-sided ideals. In commutative rings, the three concepts "left ideal", "right ideal" and "two-sided ideal" coincide. The terms "ideal" and "two-sided ideal" are used synonymously in Wikipedia.

The ring R can be considered as a left module over itself, and the left ideals of R are then seen as the submodules of this module. Similarly, the right ideals are submodules of R as a right module over itself, and the two-sided ideals are submodules of R as a bimodule over itself.

Examples

The even integers form an ideal in the ring Z of all integers; it is usually denoted by 2Z.
The set of all polynomials with real coefficients which are divisible by the polynomial x² + 1 is an ideal in the ring of all polynomials.
The set of all n-by-n matrices whose last column is zero forms a left-sided ideal in the ring of all n-by-n matrices. It is not a right-sided ideal.
The ring C(R) of all continuous functions f: R → R contains the ideal of all continuous functions f such that f(1) = 0. Another ideal in C(R) is given by those functions which vanish for large enough arguments, i.e. those continuous functions f for which there exists a number L > 0 such that f(x) = 0 whenever |x| > L.
{0} and R are ideals in every ring R. If R is commutative, then R is a field iff it has precisely two ideals, {0} and R.

Types of ideals

The first two examples above are principal ideals; the principal (left) ideal generated by an element a in R is Ra := {ra : r ∈ R}. The principal right ideal aR is defined similarly; and these two principal ideals generated by a are identical (and hence a two-sided ideal) if the ring is commutative. In that case, it's common to denote the principal ideal by <a>.

An ideal I is called proper if I ≠ R. An ideal is proper if and only if it doesn't contain 1. A proper ideal is called maximal if the only proper ideal it is contained in is itself. Every ideal is contained in a maximal ideal, a consequence of Zorn's lemma. A proper ideal I is called prime if, whenever ab belongs to I, then so does a or b (or both). Every maximal ideal is prime.

Factor rings and kernels

Ideals are important because they appear as the kernels of ring homomorphisms and allow one to define factor rings, as will be described next.

Recall that a function f: R → S is a ring homomorphism iff f(a + b) = f(a) + f(b) and f(ab) = f(a) f(b) for all a, b in R and f(1) = 1. Then the kernel of f is defined as

ker(f) := {a ∈ R : f(a) = 0}.

The kernel is always a two-sided ideal of R.

Conversely, if we start with a two-sided ideal I of R, then we may define a congruence relation ~ on R as follows: a ~ b if and only if b - a is in I. In case a ~ b, we say that a and b are congruent modulo I. The equivalence class of the element a in R is given by

[a] = a + R := {a + r : r ∈ R}.

The set of all such equivalence classes is denoted by R/I; it becomes a ring, the factor ring of R modulo I, if one defines

(a + R) + (b + R) = (a + b) + R;
(a + R) * (b + R) = (ab) + R.

The map p: R → R/I defined by p(a) = a + R is a surjective ring homomorphism (or regular epimorphism) whose kernel is the original ideal I.

In summary, we see that ideals are precisely the kernels of ring homomorphisms.

If R is commutative and I is a maximal ideal, then the factor ring R/I is a field; if I is only a prime ideal, then R/I is only an integral domain.

Ideal operations

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 lattice.

If A is any subset of the ring R, then we can define the ideal generated by A to be the smallest ideal of R containing A; it is denoted by <A> and contains all finite sums of the form ∑ r_ia_is_i with r_i and s_i in R and a_i in A. The principal ideals mentioned above are the special case when A is just the singleton {a}.

The product of two ideals I and J is defined to be the ideal IJ generated by all products of the form ab with a in I and b in J. It is contained in the intersection of I and J.

Important properties of these ideal operations are recorded in the Noether isomorphism theorems.

Ideals as "ideal numbers"

The term "ideal" comes from "ideal number": ideals were seen as a generalization of the concept of number. In the ring Z of integers, every ideal can be generated by a single number (so Z is a principal ideal domain), and the ideal determines the number up to its sign. The concepts of "ideal" and "number" are therefore almost identical in Z (and in any principal ideal domain). In other rings, it turned out that the concept of "ideal" allows one to generalize several properties of numbers. For instance, in general rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalization of coprime numbers, and one can prove a generalized Chinese remainder theorem about ideals. In a certain class of rings important in number theory, the Dedekind domains, one can even recover a version of the fundamental theorem of arithmetic: in these rings, every proper ideal can be uniquely written as a product of prime ideals.

See also: