Ideal (ring theory)

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History

Definitions and motivation

Examples and properties

Types of ideals

Ideal operations

Examples of ideal operations

Radical of a ring

Extension and contraction of an ideal

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 (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

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.^[2]^[3] Later the concept was expanded by David Hilbert and especially Emmy Noether.

Definitions and motivation

For an arbitrary ring, letbe itsadditive group. A subsetis called a left ideal ofif it is an additive subgroup ofthat "absorbs multiplication from the left by elements of"; that is,is a left ideal if it satisfies the following two conditions:

is a subgroup of
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 (resp. right, two-sided) ideal of R as a left (resp. right, bi-) 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 (note that ℤ is a commutative ring). The key observation here is that we obtain ℤn by taking ℤ and wrapping it around itself so that n gets identified with 0 (since n is congruent to 0 modulo n). However, when doing so we must ensure that the resulting structure is again a ring. This requirement forces us to make some 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 the set nℤ = {nm | m∈ℤ} 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 = { x r : r ∈ R }. 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 (S) such that S ⊆ (S). The ring that we obtain after the identification depends only on the ideal (S) and not on the set S that we started with. That is, if (S) = (T), 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:

I is an additive subgroup of R: the zero 0 of R is a "zero" 0 ∈ I, and if x1∈Iandx2 ∈ I are "zeros", then x1*-x2* ∈ I is a "zero" too.
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. (In fact, no other elements should be designated as "zero" if we want to make the least amount of identifications.)

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.

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 .^[1] 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).^[4] 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.)^[5] 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 x − y ∈ I. 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

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.^[6]
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 a**n 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 b**n 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.
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.
Parameter ideal: an ideal generated by a system of parameters.

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

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

which is a left (resp. right) ideal, and, ifare two-sided,

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

Noteis the smallest left (resp. right) ideal containing bothand(or the union), while the productis contained in the intersection ofand.

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 ifcontainsor.

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.

Ifare ideals of a commutative ring R, thenin 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 theTor functor:^[7])

An integral domain is called aDedekind domainif for each pair of ideals, there is an idealsuch that.^[8] 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 thefundamental theorem of arithmetic.

Examples of ideal operations

Inwe have

sinceis the set of integers which are divisible by bothand.

Letand 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.^[9]^[10]^[11]

Radical of a ring

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, aprimitive idealof R is the annihilator of a (nonzero)simple R-module. TheJacobson radicalof R is the intersection of all primitive ideals. Equivalently,

Indeed, ifis a simple module and x is a nonzero element in M, thenand, meaningis a maximal ideal. Conversely, ifis a maximal ideal, thenis 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 thatis aunit elementif and only ifis (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 amaximal submodule, since if there is a maximal submodule,and so, a contradiction. Since a nonzerofinitely generated moduleadmits a maximal submodule, in particular, one has:

Ifand M is finitely generated, then

A maximal ideal is a prime ideal and so one has

where the intersection on the left is called thenilradicalof R. As it turns out,is also the set ofnilpotent elementsof R.

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

Extension and contraction of an ideal

Let A and B be twocommutative rings, and let f : A → B be aring homomorphism. Ifis an ideal in A, thenneed not be an ideal in B (e.g. take f to be theinclusionof the ring of integers Z into the field of rationals Q). The extensionofin B is defined to be the ideal in B generated by. Explicitly,

Ifis an ideal of B, thenis always an ideal of A, called the contractionofto A.

Assuming f : A → B 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, thatbeing prime (or maximal) in A implies thatis prime (or maximal) in B. Many classic examples of this stem from algebraic number theory. For example,embedding. In, the element 2 factors aswhere (one can show) neither ofare units in B. Sois not prime in B (and therefore not maximal, as well). Indeed,shows that,, and therefore.

On the other hand, if f issurjectiveandthen:

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 afield extensionof L, and let B and A be therings of integersof K and L, respectively. Then B is anintegral extensionof A, and we let f be theinclusion mapfrom A to B. The behaviour of aprime idealof A under extension is one of the central problems ofalgebraic number theory.

The following is sometimes useful:^[12] a prime idealis a contraction of a prime ideal if and only if. (Proof: Assuming the latter, noteintersects, a contradiction. Now, the prime ideals ofcorrespond to those in B that are disjoint from. Hence, there is a prime idealof B, disjoint from, such thatis a maximal ideal containing. One then checks thatlies over. The converse is obvious.)

References

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Citation Linkopenlibrary.orgSome authors call the zero and unit ideals of a ring R the trivial ideals of R.

Sep 25, 2019, 11:33 PM

[2]

Citation Linkopenlibrary.orgHarold M. Edwards (1977). Fermat's last theorem. A genetic introduction to algebraic number theory. p. 76.

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Citation Linkopenlibrary.orgEverest G., Ward T. (2005). An introduction to number theory. p. 83.

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Citation Linkopenlibrary.orgLang, Serge (2005). Undergraduate Algebra (Third ed.). Springer-Verlag. ISBN 978-0-387-22025-3, Section III.2

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[5]

Citation Linkopenlibrary.orgIf 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.

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[6]

Citation Linkbooks.google.comBecause simple commutative rings are fields. See Lam (2001). A First Course in Noncommutative Rings. p. 39.

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[7]

Citation Linkopenlibrary.org, Exercise A 3.17

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[8]

Citation Linkopenlibrary.org, page 9.

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[9]

Citation Linkwww.math.uiuc.edu"ideals". www.math.uiuc.edu. Retrieved 2017-01-14.