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 (even or odd) 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 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 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.
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 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.
Notes on convention Some authors do not require a ring to have the multiplicative identity; for these authors, the term "ring" refers to what others call a "rng". For a rng , a left ideal is a with the additional property that is in for every and every . (Right and two-sided ideals are defined similarly.) For a ring, an ideal (say a left ideal) is rarely a subring; since a subring shares the same multiplicative identity with the ambient ring , if were a subring, for every , we have i.e., .
The notion of an ideal does not involve associativity; thus, an ideal is also defined for non-associative rings.
For algebras, we additionally assume that an ideal is a linear subspace. If a -algebra is unital, then any ring-ideal is an algebra-ideal, since when and . For non-unital algebras, there can exist ring-ideals that are not algebra-ideals, for example if for all , then ring-ideals of are subrings of , while algebra-ideals of are linear subspaces of .
Traditionally, ideals are denoted using Fraktur lower-case letters, generally the first few letters (, etc.) for generic ideals, for maximal ideals, and for prime ideals. In modern texts, capital letters, like or (or and for maximal and prime ideals, respectively) are also commonly used.
Examples and properties (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.) 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 (see below) 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 .Some authors call the zero and unit ideals of a ring R the trivial ideals of R. Every (left, right or two-sided) ideal contains the zero ideal and is contained in the unit ideal. An (left, right or two-sided) ideal that is not the unit ideal is called a proper ideal (as it is a proper subset). 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 (or right) ideals. (Proof: if is a nonzero element, then the principal left ideal (see below) is nonzero and thus ; i.e., for some nonzero . Likewise, for some nonzero . Then .) 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 (so ), 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 (resp. right, two-sided) ideals of containing the kernel of and the left (resp. right, two-sided) 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 (see the Types of ideals section for the definitions of these 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. (Note: this fact remains true even if R is without the unity 1.) 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. (Again this is still valid if the ring R lacks the unity 1.) When , taking and , in particular, there exists a left ideal that is maximal among proper left ideals (often simply called a maximal left ideal); see Krull's theorem for more. A left (resp. right, two-sided) ideal generated by a single element x is called the principal left (resp. right, two-sided) ideal generated by x and is denoted by (resp. ). 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 (finite) left R-linear combinations of elements of X over R: (since such a span is the smallest left ideal containing X.)If R does not have a unit, then the internal descriptions above must be modified slightly. In addition to the finite sums of products of things in X with things in R, we must allow the addition of n-fold sums of the form , and n-fold sums of the form for every x in X and every n in the natural numbers. When R has a unit, this extra requirement becomes superfluous. A right (resp. two-sided) 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 (finite or infinite) set of indexed ring elements is denoted or . There is a bijective correspondence between ideals and congruence relations (equivalence relations that respect the ring structure) 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.Because simple commutative rings are fields. See Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal. Zero ideal: the ideal . Unit ideal: the whole ring (being the ideal generated by ). 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.