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Wikipedia / 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 (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 that is a subgroup of the additive group that is closed under left multiplication by elements of and every , one has

  • .

In other words, a left ideal is a left submodule, considered as a left module over itself.

A right ideal 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".

is an abelian subgroup, the relation between and defined by

is an equivalence relation, and the set of equivalence classes is an abelian group denoted and called the quotient. is a left or a right ideal, the quotient is a left or right -module, respectively.

is a ring, and the function

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". , a left ideal is a .

The notion of an ideal does not involve associativity; thus, an ideal is also defined for non-associative rings.

.

Traditionally, ideals are denoted using Fraktur 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 since it is precisely the two-sided ideal generated (see below) by the unity consisting of only the additive identity 0R forms a two-sided ideal called the zero ideal and is denoted by . 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). for every . Typically there are plenty of proper ideals. In fact, if R is a skew-field are its only ideals and conversely: that is, a nonzero ring R for some nonzero .)

  • The even integers 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 is an ideal denoted is generated by its smallest positive element, as a consequence of Euclidean division is a principal ideal domain.

  • The set of all polynomials is an ideal in the ring of all real-coefficient polynomials .

  • and positive integer . For each matrices matrices with entries in . It is not a left ideal. Similarly, for each -th column is zero is a left ideal but not a right ideal.

  • of all continuous functions under pointwise multiplication 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.

  • is a ring homomorphism is a two-sided ideal of . By definition, is not the zero ring (so is a proper ideal. More generally, for each left ideal I of S is a left ideal. If I is a left ideal of R of S: unless f need not be an ideal of S; see also .

  • Ideal correspondence: Given a surjective ring homomorphism 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 a subset, then the annihilator of S of a commutative ring R, the R is an ideal of R called the ideal quotient and is denoted by ; it is an instance of idealizer in commutative algebra.

  • be an ascending chain of left ideals in a ring R for each 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 is a left ideal that is disjoint from E and disjoint from E. (Again this is still valid if the ring R 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 (resp. is often also denoted by .

  • 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 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.) 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., is a finite set is also written as .

  • There is a bijective correspondence between ideals and congruence relations of a ring if is a congruence relation on on , let 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.

Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details:

  • Fractional ideal is a commutative domain with quotient field-submodule .

  • 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.

Ideal operations

and , left (resp. right) ideals of a ring R, their sum is

are two-sided,

i.e. the product is the ideal generated by all products of the form ab with a and b in .

(or the union and .

The distributive law holds for two-sided ideals ,

  • ,

  • .

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.

are ideals of a commutative ring R in the following two cases (at least)

(More generally, the difference between a product and an intersection of ideals is measured by the Tor functor: .)

An integral domain is called a Dedekind domain 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

we have

and .

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.

  1. For simplicity, we work with commutative rings but, with some changes, the results are also true for non-commutative rings.

Let R be a commutative ring. By definition, a primitive ideal of R is the annihilator of a (nonzero) simple R-module. The Jacobson radical of R is the intersection of all primitive ideals. Equivalently,

is a simple module and x is a nonzero element in M is the annihilator of the simple R-module . There is also another characterization (the proof is not hard):

is a unit element is (see the link) 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 (Nakayama's lemma) 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:

  1. and M is finitely generated, then .

A maximal ideal is a prime ideal and so one has

where the intersection on the left is called the nilradical of R is also the set of nilpotent elements of R.

If R is an Artinian ring is nilpotent and for some n. That is, , a contradiction.)

Extension and contraction of an ideal

be two commutative rings be a ring homomorphism to be the inclusion). The extension generated by . Explicitly,

is another common notation for this ideal extension.

, called the contraction.

, then:

  • ,

. Many classic examples of this stem from algebraic number theory. For example, consider the embedding shows that , , and therefore .

is surjective and \mathfrak{a} \supseteq \ker f then:

Remark be a field extension be the rings of integers is an integral extension be the inclusion map. The behaviour of a prime ideal under extension is one of the central problems of algebraic number theory.

The following is sometimes useful:. The converse is obvious.)

Generalizations

Ideals can be generalized to any monoid object is the object where the monoid structure has been forgotten. A left ideal is a subobject that "absorbs multiplication from the left by elements of is a left ideal if it satisfies the following two conditions:

  1. is a subobject

  2. and every is in .

A right ideal is defined with the condition "" replaced by "'". A two-sided ideal is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.

See also

Notes

References