Ideal (ring theory)


In ring theory,[] a branch of abstract algebra, an ideal is a special subset of a ring. 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 any other integer results in another even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring similarly to the way that, 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 be distinct from 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.
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.

History

Ideals were first proposed by Richard Dedekind in 1876 in the third edition of his book Vorlesungen über Zahlentheorie. They were a generalization of the concept of ideal numbers developed by Ernst Kummer. Later the concept was expanded by David Hilbert and especially Emmy Noether.

Definitions and motivation

For an arbitrary ring, let be its additive group. A subset is called a left ideal of if it is an additive subgroup of that "absorbs multiplication from the left by elements of "; that is, is a left ideal if it satisfies the following two conditions:
  1. is a subgroup of
  2. For every and every, the product is in.
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.
We can view a left ideal of R as a left R-submodule of R 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 n must be identified with 0 since n is congruent to 0 modulo n, and 2) the resulting structure must again be a ring. The second requirement forces us to make additional identifications. 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?
The answer is, unsurprisingly, the set nℤ = of all integers congruent to 0 modulo n. That is, we must wrap ℤ around itself infinitely many times so that the integers..., n ⋅ -2, n ⋅ -1, n ⋅ +1, n ⋅ +2,... will all align with 0. If we look at what properties this set must satisfy in order to ensure that ℤn is a ring, then we arrive at the definition of an ideal. Indeed, one can directly verify that nℤ is an ideal of ℤ.
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.
We can make a similar construction in any commutative ring R: start with an arbitrary x ∈ R, and then identify with 0 all elements of the ideal xR =. It turns out that the ideal xR is the smallest ideal that contains x, called the ideal generated by x. More generally, we can start with an arbitrary subset S ⊆ R, and then identify with 0 all the elements in the ideal generated by S: the smallest ideal such that S ⊆. The ring that we obtain after the identification depends only on the ideal and not on the set S that we started with. That is, if =, then the resulting rings will be the same.
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 "zeroes" by R/I:
  1. I is an additive subgroup of R: the zero 0 of R is a "zero" 0 ∈ I, and if x1 ∈ I and x2 ∈ I are "zeros", then x1 - x2 ∈ I is a "zero" too.
  2. Any r ∈ R multiplied by a "zero" x ∈ I is a "zero" rx ∈ I.
It turns out that the above conditions are also sufficient for I to contain all the necessary "zeroes": no other elements have to be designated as "zero" in order to form R/I.
Remark. If R is not necessarily commutative, the above construction still works using two-sided ideals.

Examples and properties

For the sake of succinctness, 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:
The sum and product of ideals are defined as follows. For and, left ideals of a ring R, their sum is
which is a left ideal,
and, if are two-sided,
i.e. the product is the ideal generated by all products of the form ab with a in and b in.
Note is the smallest left ideal containing both and , while the product is contained in the intersection of and.
The distributive law holds for two-sided ideals,
If a product is replaced by an intersection, a partial distributive law holds:
where the equality holds if contains or.
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, and product make the set of ideals of a commutative ring into a quantale.
If are ideals of a commutative ring R, then in the following two cases
An integral domain is called a Dedekind domain if for each pair of ideals, there is an ideal such that. 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.

Examples of ideal operations

In we have
since is the set of integers which are divisible by both and.
Let and let. Then,
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.

Radical of a ring

Ideals appear naturally in the study of modules, especially in the form of a radical.
Let R be a commutative ring. By definition, a primitive ideal of R is the annihilator of a simple R-module. The Jacobson radical of R is the intersection of all primitive ideals. Equivalently,
Indeed, if is a simple module and x is a nonzero element in M, then and, meaning is a maximal ideal. Conversely, if is a maximal ideal, then is the annihilator of the simple R-module. There is also another characterization :
For a not-necessarily-commutative ring, it is a general fact that is a unit element if and only if is 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 is built-in to the definition of a Jacobson radical: if M is a module such that, then M does not admit a maximal submodule, since if there is a maximal submodule, and so, a contradiction. 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 is nilpotent and.

Extension and contraction of an ideal

Let A and B be two commutative rings, and let f : A → B be a ring homomorphism. If is an ideal in A, then need not be an ideal in B. The extension of in B is defined to be the ideal in B generated by. Explicitly,
If is an ideal of B, then is always an ideal of A, called the contraction of to A.
Assuming f : A → B is a ring homomorphism, is an ideal in A, is an ideal in B, then:
It is false, in general, that being prime in A implies that is prime in B. Many classic examples of this stem from algebraic number theory. For example, embedding. In, the element 2 factors as where neither of are units in B. So is not prime in B. Indeed, shows that,, and therefore.
On the other hand, if f is surjective and kernel_| 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. Then B is an integral extension of A, and we let f be the inclusion map from A to B. The behaviour of a prime ideal of A under extension is one of the central problems of algebraic number theory.
The following is sometimes useful: a prime ideal is a contraction of a prime ideal if and only if.

Generalisations

Ideals can be generalised to any monoid object, where is the object where the monoid structure has been forgotten. A left ideal of is a subobject that "absorbs multiplication from the left by elements of "; that is, is a left ideal if it satisfies the following two conditions:
  1. is a subobject of
  2. For every and every, the product is in.
A right ideal is defined with the condition "" replaced by "'". A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. When is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.