Ideal (ring theory)

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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 (a type of ring important in number theory).

The concept of an order 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[edit]

Ideals were first proposed by Richard Dedekind in 1876 in the third edition of his book Vorlesungen über Zahlentheorie (English: Lectures on Number Theory). They were a generalization of the concept of ideal numbers developed by Ernst Kummer.[1][2] Later the concept was expanded by David Hilbert and especially Emmy Noether.

Definitions and motivation[edit]

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 "" 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 ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.

Ideals of a ring are in some sense a generalisation of the set of multiples of an integer. That is, . Intuitively, the definition can be motivated as follows: Suppose we have a subset of a ring and that we would like to obtain a ring with the same structure as , except that the elements of should be zero (they are in some sense "negligible").

But if and in our new ring, then surely should be zero too, and as well as should be zero for any element (zero or not).

The definition of an ideal is such that the ideal generated (see below) by is exactly the set of elements that are forced to become zero if becomes zero, and the quotient ring is the desired ring where is zero, and only elements that are forced by to be zero are zero. The requirement that and should have the same structure (except that becomes zero) is formalized by the condition that the projection from to is a (surjective) ring homomorphism.

The quintessential example is when and for some nonzero . By "considering to be zero", we are essentially reducing elements of to their remainder after dividing by n. Thus and is the set of possible remainders after dividing an integer by n (ie. the cyclic group of order n).

In the language of module theory, a left (resp. right, two-sided) ideal is precisely a left (resp. right, bi-) -submodule of . That is to say, a left ideal, for example, is an additive subgroup that is stable under the left multiplications by the elements of R.

Examples and properties[edit]

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.

  • 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 1. Also, the set consisting of only the additive identity 0R forms a two-sided ideal called the zero ideal and is denoted by . These two ideals are referred to as the trivial ideals of R. Every (left, right or two-sided) ideal contains the zero ideal and is contained in the unit ideal.
  • A (left, right or two-sided) ideal that is not the unit ideal is called a proper ideal (as it is a proper subset).[3] 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; it is usually denoted by . This is because the sum of any even integers is even, and the product of any integer with an even integer is also even. Similarly, the set of all integers divisible by a fixed integer n is an ideal denoted .
  • 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 row is zero forms a right ideal in the ring of all n-by-n matrices. It is not a left ideal. The set of all n-by-n matrices whose last column is zero forms a left ideal but not a right ideal.
  • The ring of all continuous functions f from to under pointwise multiplication contains the ideal of all continuous functions f such that f(1) = 0. Another ideal in 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.
  • 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 convention, and thus if , 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 #Extension and contraction of an ideal below.
  • 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 .
  • (For those who know modules) 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.
  • 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.)[4] 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.,
  • 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 . If is a finite set, then is also written as .
  • In the ring of integers, every ideal can be generated by a single number (so is a principal ideal domain), as a consequence of Euclidean division (or some other way).
  • There is a bijective correspondence between ideals and congruence relations (equivalence relations that respect the ring structure) on the ring: Given an ideal I of a ring R, let x ~ y if xyI. Then ~ is a congruence relation on R. Conversely, given a congruence relation ~ on R, let I = {x : x ~ 0}. Then I is an ideal of R.

Types of ideals[edit]

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 I is called a maximal ideal if there exists no other proper ideal J with I a proper subset of J. The factor ring of a maximal ideal is a simple ring in general and is a field for commutative rings.[5]
  • Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal.
  • Prime ideal: A proper ideal I is called a prime ideal if for any a and b in R, if ab is in I, then at least one of a and b is in I. 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 I is called radical or semiprime if for any a in R, if an is in I for some n, then a is in I. 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 I is called a primary ideal if for all a and b in R, if ab is in I, then at least one of a and bn is in I for some natural number n. 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. A right primitive ideal is defined similarly. Actually (despite the name) the left and right primitive ideals are always two-sided ideals. Primitive ideals are prime. A factor rings constructed with a right (left) primitive ideals is a right (left) primitive ring. For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields.
  • Irreducible ideal: An ideal is said to be irreducible if it cannot be written as an intersection of ideals which 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.

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 R is a commutative domain with quotient field K. Despite their names, fractional ideals are R submodules of K with a special property. If the fractional ideal is contained entirely in R, then it is truly an ideal of R.
  • Invertible ideal: Usually an invertible ideal A is defined as a fractional ideal for which there is another fractional ideal B such that AB=BA=R. Some authors may also apply "invertible ideal" to ordinary ring ideals A and B with AB=BA=R in rings other than domains.

Ideal operations[edit]

The sum and product of ideals are defined as follows. For and , left (resp. right) ideals of a ring R, their sum is

,

which is a left (resp. right) 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 (resp. right) ideal containing both and (or the union ), 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 (or join), 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 (at least)

  • is generated by elements that form a regular sequence modulo .

(More generally, the difference between a product and an intersection of ideals is measured by the Tor functor:[6])

An integral domain is called a Dedekind domain if for each pair of ideals , there is an ideal such that .[7] 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[edit]

In we have

since is the set of integers which are divisible by both and .

Let and let . Then,

  • and
  • while

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.[8][9][10]

Radical of a ring[edit]

Ideals appear naturally in the study of modules, especially in the form of a radical.

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,

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 (the proof is not hard):

For a not-necessarily-commutative ring, it is a general fact that is a unit element if and only if 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:

If 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. As it turns out, is also the set of nilpotent elements of R.

If R is an Artinian ring, then is nilpotent and . (Proof: first note the DCC implies for some n. If (DCC) is an ideal properly minimal over the latter, then . That is, , a contradiction.)

Extension and contraction of an ideal[edit]

Let A and B be two commutative rings, and let f : AB be a ring homomorphism. If is an ideal in A, then need not be an ideal in B (e.g. take f to be the inclusion of the ring of integers Z into the field of rationals Q). 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 : AB is a ring homomorphism, is an ideal in A, is an ideal in B, then:

  • is prime in B is prime in A.

It is false, in general, that being prime (or maximal) in A implies that is prime (or maximal) in B. Many classic examples of this stem from algebraic number theory. For example, embedding . In , the element 2 factors as where (one can show) neither of are units in B. So is not prime in B (and therefore not maximal, as well). Indeed, shows that , , and therefore .

On the other hand, if f is surjective and then:

  • and .
  • is a prime ideal in A is a prime ideal in B.
  • is a maximal ideal in A is a maximal ideal in B.

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:[11] a prime ideal is a contraction of a prime ideal if and only if . (Proof: Assuming the latter, note intersects , a contradiction. Now, the prime ideals of correspond to those in B that are disjoint from . Hence, there is a prime ideal of B, disjoint from , such that is a maximal ideal containing . One then checks that lies over . The converse is obvious.)

See also[edit]

References[edit]

  1. ^ Harold M. Edwards (1977). Fermat's last theorem. A genetic introduction to algebraic number theory. p. 76.
  2. ^ Everest G., Ward T. (2005). An introduction to number theory. p. 83.
  3. ^ Lang 2005, Section III.2
  4. ^ 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 x+x+...+x, and n-fold sums of the form (−x)+(−x)+...+(−x) for every x in X and every n in the natural numbers. When R has a unit, this extra requirement becomes superfluous.
  5. ^ Because simple commutative rings are fields. See Lam (2001). A First Course in Noncommutative Rings. p. 39.
  6. ^ Eisenbud, Exercise A 3.17
  7. ^ Milnor, page 9.
  8. ^ "ideals". www.math.uiuc.edu. Retrieved 2017-01-14.
  9. ^ "sums, products, and powers of ideals". www.math.uiuc.edu. Retrieved 2017-01-14.
  10. ^ "intersection of ideals". www.math.uiuc.edu. Retrieved 2017-01-14.
  11. ^ Atiyah–MacDonald, Proposition 3.16.

External links[edit]