Ideal (ring theory)


In mathematics, and more specifically in ring theory, 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 any 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.
The related, but distinct, concept of an ideal in order theory is derived from the notion of an 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

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.

Definitions

Given a ring, a left ideal is a subset of that is a subgroup of the additive group of that is closed under left multiplication by elements of ; that is, and for every and every, one has
  • .
In other words, a left ideal is a left submodule of, considered as a left module over itself.
A right ideal is defined similarly, with the condition replaced by. A two-sided ideal is a left ideal that is also a right ideal.
If the ring is commutative, the definitions of left, right, and two-sided ideal coincide, and one talks simply of an ideal. In the non-commutative case, "ideal" is often used instead of "two-sided ideal".
Since an ideal is an abelian subgroup, the relation between and defined by
is an equivalence relation on, and the set of equivalence classes is an abelian group denoted and called the quotient of by. If is a left or a right ideal, the quotient is a left or right -module, respectively.
If the ideal is two-sided, the quotient is a ring, and the function
that associates to each element of its equivalence class is a surjective ring homomorphism that has the ideal as its kernel. Conversely, the kernel of a ring homomorphism is a two-sided ideal. Therefore, ''the two-sided ideals are exactly the kernels of ring homomorphisms.''

Examples and properties

  • In a ring R, the set R itself forms a two-sided ideal of R called the unit ideal. It is often also denoted by since it is precisely the two-sided ideal generated by the unity. Also, the set consisting of only the additive identity 0R forms a two-sided ideal called the zero ideal and is denoted by. Every ideal contains the zero ideal and is contained in the unit ideal.
  • An ideal that is not the unit ideal is called a proper ideal. Note: a left ideal is proper if and only if it does not contain a unit element, since if is a unit element, then for every. Typically there are plenty of proper ideals. In fact, if R is a skew-field, then are its only ideals and conversely: that is, a nonzero ring R is a skew-field if are the only left ideals.
  • The even integers form an ideal in the ring of all integers, since the sum of any two even integers is even, and the product of any integer with an even integer is also even; this ideal is usually denoted by. More generally, the set of all integers divisible by a fixed integer is an ideal denoted. In fact, every non-zero ideal of the ring is generated by its smallest positive element, as a consequence of Euclidean division, so is a principal ideal domain.
  • The set of all polynomials with real coefficients that are divisible by the polynomial is an ideal in the ring of all real-coefficient polynomials.
  • Take a ring and positive integer. For each, the set of all matrices with entries in whose -th row is zero is a right ideal in the ring of all matrices with entries in. It is not a left ideal. Similarly, for each, the set of all matrices whose -th column is zero is a left ideal but not a right ideal.
  • The ring of all continuous functions from to under pointwise multiplication contains the ideal of all continuous functions such that. Another ideal in is given by those functions that vanish for large enough arguments, i.e. those continuous functions for which there exists a number such that whenever.
  • A ring is called a simple ring if it is nonzero and has no two-sided ideals other than. Thus, a skew-field is simple and a simple commutative ring is a field. The matrix ring over a skew-field is a simple ring.
  • If is a ring homomorphism, then the kernel is a two-sided ideal of. By definition,, and thus if is not the zero ring, then is a proper ideal. More generally, for each left ideal I of S, the pre-image is a left ideal. If I is a left ideal of R, then is a left ideal of the subring of S: unless f is surjective, need not be an ideal of S; see also.
  • Ideal correspondence: Given a surjective ring homomorphism, there is a bijective order-preserving correspondence between the left ideals of containing the kernel of and the left ideals of : the correspondence is given by and the pre-image. Moreover, for commutative rings, this bijective correspondence restricts to prime ideals, maximal ideals, and radical ideals.
  • If M is a left R-module and a subset, then the annihilator of S is a left ideal. Given ideals of a commutative ring R, the R-annihilator of is an ideal of R called the ideal quotient of by and is denoted by ; it is an instance of idealizer in commutative algebra.
  • Let be an ascending chain of left ideals in a ring R; i.e., is a totally ordered set and for each. Then the union is a left ideal of R.
  • The above fact together with Zorn's lemma proves the following: if is a possibly empty subset and is a left ideal that is disjoint from E, then there is an ideal that is maximal among the ideals containing and disjoint from E. When, taking and, in particular, there exists a left ideal that is maximal among proper left ideals ; see Krull's theorem for more.
  • A left ideal generated by a single element x is called the principal left ideal generated by x and is denoted by . The principal two-sided ideal is often also denoted by or.
  • An arbitrary union of ideals need not be an ideal, but the following is still true: given a possibly empty subset X of R, there is the smallest left ideal containing X, called the left ideal generated by X and is denoted by. Such an ideal exists since it is the intersection of all left ideals containing X. Equivalently, is the set of all the left R-linear combinations of elements of X over R: A right ideal generated by X is defined in the similar way. For "two-sided", one has to use linear combinations from both sides; i.e., If is a finite set, then is also written as or. More generally, the two-sided ideal generated by a set of indexed ring elements is denoted or.
  • There is a bijective correspondence between ideals and congruence relations on the ring: Given an ideal of a ring, let if. Then is a congruence relation on. Conversely, given a congruence relation on, let. Then is an ideal of.

    Types of ideals

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.
  • Maximal ideal: A proper ideal is called a maximal ideal if there exists no other proper ideal with a proper subset of. The factor ring of a maximal ideal is a simple ring in general and is a field for commutative rings.
  • Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal.
  • Zero ideal: the ideal.
  • Unit ideal: the whole ring.
  • Prime ideal: A proper ideal is called a prime ideal if for any and in, if is in, then at least one of and is in. The factor ring of a prime ideal is a prime ring in general and is an integral domain for commutative rings.
  • Radical ideal or semiprime ideal: A proper ideal is called radical or semiprime if for any in, if is in for some, then is in. The factor ring of a radical ideal is a semiprime ring for general rings, and is a reduced ring for commutative rings.
  • Primary ideal: An ideal is called a primary ideal if for all and in, if is in, then at least one of and is in for some natural number. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime.
  • Principal ideal: An ideal generated by one element.
  • Finitely generated ideal: This type of ideal is finitely generated as a module.
  • Primitive ideal: A left primitive ideal is the annihilator of a simple left module.
  • Irreducible ideal: An ideal is said to be irreducible if it cannot be written as an intersection of ideals that properly contain it.
  • Comaximal ideals: Two ideals, are said to be comaximal if for some and.
  • Regular ideal: This term has multiple uses. See the article for a list.
  • Nil ideal: An ideal is a nil ideal if each of its elements is nilpotent.
  • Nilpotent ideal: Some power of it is zero.
  • Parameter ideal: an ideal generated by a system of parameters.
  • Perfect ideal: A proper ideal in a Noetherian ring is called a perfect ideal if its grade equals the projective dimension of the associated quotient ring,. A perfect ideal is unmixed.
  • Unmixed ideal: A proper ideal in a Noetherian ring is called an unmixed ideal if the height of is equal to the height of every associated prime of. (This is stronger than saying that is equidimensional. See also equidimensional ring.
Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details:
  • Fractional ideal: This is usually defined when is a commutative domain with quotient field. Despite their names, fractional ideals are not necessarily ideals. A fractional ideal of is an -submodule of for which there exists a non-zero such that. If the fractional ideal is contained entirely in, then it is truly an ideal of.
  • Invertible ideal: Usually an invertible ideal is defined as a fractional ideal for which there is another fractional ideal such that. Some authors may also apply "invertible ideal" to ordinary ring ideals and with in rings other than domains.