Ideal (ring theory) explained

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 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.[1] 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.[2] [3] 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

R

that "absorbs multiplication from the left by elements of "; that is,

I

is a left ideal if it satisfies the following two conditions:

(I,+)

is a subgroup of,
  1. For every

r\inR

and every, the product

rx

is in .

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

rx\inI

replaced by . A two-sided ideal is a left ideal that is also a right ideal.

If the ring is commutative, the three definitions are the same, and one talks simply of an ideal. In the non-commutative case, "ideal" is often used instead of "two-sided ideal".

If is a left, right or two-sided ideal, the relation

x-y\inI

is an equivalence relation on, and the set of equivalence classes forms a left, right or two-sided module denoted

R/I

and called the quotient of by . (This is a generalization of modular arithmetic.)

If the ideal is two-sided,

R/I

is a ring, and the function

R\toR/I

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.

Note on convention

By convention, a ring has the multiplicative identity. But some authors do not require a ring to have the multiplicative identity; i.e., for them, a ring is a rng. For a rng, a left ideal is a subrng with the additional property that

rx

is in for every

r\inR

and every

x\inI

. (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

r\inR

, we have

r=r1\inI;

i.e.,

I=R

.

The notion of an ideal does not involve associativity; thus, an ideal is also defined for non-associative rings (often without the multiplicative identity) such as a Lie algebra.

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.)

(1)

since it is precisely the two-sided ideal generated (see below) by the unity . Also, the set

\{0R\}

consisting of only the additive identity 0R forms a two-sided ideal called the zero ideal and is denoted by .[4] Every (left, right or two-sided) ideal contains the zero ideal and is contained in the unit ideal.

ak{a}

is proper if and only if it does not contain a unit element, since if

u\inak{a}

is a unit element, then

r=(ru-1)u\inak{a}

for every . Typically there are plenty of proper ideals. In fact, if R is a skew-field, then

(0),(1)

are its only ideals and conversely: that is, a nonzero ring R is a skew-field if

(0),(1)

are the only left (or right) ideals. (Proof: if

x

is a nonzero element, then the principal left ideal

Rx

(see below) is nonzero and thus

Rx=(1)

; i.e.,

yx=1

for some nonzero . Likewise,

zy=1

for some nonzero

z

. Then

z=z(yx)=(zy)x=x

.)

Z

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

n

is an ideal denoted . In fact, every non-zero ideal of the ring

Z

is generated by its smallest positive element, as a consequence of Euclidean division, so

Z

is a principal ideal domain.

x2+1

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

R

and positive integer . For each, the set of all

n x n

matrices with entries in

R

whose

i

-th row is zero is a right ideal in the ring

Mn(R)

of all

n x n

matrices with entries in . It is not a left ideal. Similarly, for each, the set of all

n x n

matrices whose

j

-th column is zero is a left ideal but not a right ideal.

C(R)

of all continuous functions

f

from

R

to

R

under pointwise multiplication contains the ideal of all continuous functions

f

such that . Another ideal in

C(R)

is given by those functions that 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 .

f:R\toS

is a ring homomorphism, then the kernel

\ker(f)=f-1(0S)

is a two-sided ideal of . By definition,, and thus if

S

is not the zero ring (so), then

\ker(f)

is a proper ideal. More generally, for each left ideal I of S, the pre-image

f-1(I)

is a left ideal. If I is a left ideal of R, then

f(I)

is a left ideal of the subring

f(R)

of S: unless f is surjective,

f(I)

need not be an ideal of S; see also
  1. Extension and contraction of an ideal
below.

R

containing the kernel of

f

and the left (resp. right, two-sided) ideals of

S

: the correspondence is given by

I\mapstof(I)

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).

S\subsetM

a subset, then the annihilator

\operatorname{Ann}R(S)=\{r\inR\midrs=0,s\inS\}

of S is a left ideal. Given ideals

ak{a},ak{b}

of a commutative ring R, the R-annihilator of

(ak{b}+ak{a})/ak{a}

is an ideal of R called the ideal quotient of

ak{a}

by

ak{b}

and is denoted by ; it is an instance of idealizer in commutative algebra.

ak{a}i,i\inS

be an ascending chain of left ideals in a ring R; i.e.,

S

is a totally ordered set and

ak{a}i\subsetak{a}j

for each . Then the union

stylecupiak{a}i

is a left ideal of R. (Note: this fact remains true even if R is without the unity 1.)

E\subsetR

is a possibly empty subset and

ak{a}0\subsetR

is a left ideal that is disjoint from E, then there is an ideal that is maximal among the ideals containing

ak{a}0

and disjoint from E. (Again this is still valid if the ring R lacks the unity 1.) When

R\ne0

, taking

ak{a}0=(0)

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.

RX

is the set of all the (finite) left R-linear combinations of elements of X over R:

RX=\{r1x1+...+rnxn\midn\inN,ri\inR,xi\inX\}.

(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.,

RXR=\{r1x1s1+...+rnxnsn\midn\inN,ri\inR,si\inR,xi\inX\}.

Rx

(resp.). The principal two-sided ideal

RxR

is often also denoted by . If

X=\{x1,...,xn\}

is a finite set, then

RXR

is also written as .

I

of a ring, let

x\simy

if . Then

\sim

is a congruence relation on . Conversely, given a congruence relation

\sim

on, let . Then

I

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.

I

is called a prime ideal if for any

a

and

b

in, if

ab

is in, then at least one of

a

and

b

is in . The factor ring of a prime ideal is a prime ring in general and is an integral domain for commutative rings.

x+y=1

for some

x\inI

and .

R

is called a perfect ideal if its grade equals the projective dimension of the associated quotient ring,[7] . A perfect ideal is unmixed.

R

is called an unmixed ideal (in height) if the height of I is equal to the height of every associated prime P of R/I. (This is stronger than saying that R/I 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:

Ideal operations

The sum and product of ideals are defined as follows. For

ak{a}

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

ak{a}+ak{b}:=\{a+b\mida\inak{a}andb\inak{b}\}

,which is a left (resp. right) ideal,and, if

ak{a},ak{b}

are two-sided,

ak{a}ak{b}:=\{a1b1+...+anbn\midai\inak{a}andbi\inak{b},i=1,2,...,n;forn=1,2,...\},

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

ak{a}

and b in .

Note

ak{a}+ak{b}

is the smallest left (resp. right) ideal containing both

ak{a}

and

ak{b}

(or the union), while the product

ak{a}ak{b}

is contained in the intersection of

ak{a}

and .

The distributive law holds for two-sided ideals,

If a product is replaced by an intersection, a partial distributive law holds:

ak{a}\cap(ak{b}+ak{c})\supsetak{a}\capak{b}+ak{a}\capak{c}

where the equality holds if

ak{a}

contains

ak{b}

or

ak{c}

.

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

ak{a},ak{b}

are ideals of a commutative ring R, then

ak{a}\capak{b}=ak{a}ak{b}

in the following two cases (at least)

ak{a}+ak{b}=(1)

ak{a}

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: .)

An integral domain is called a Dedekind domain if for each pair of ideals

ak{a}\subsetak{b}

, there is an ideal

ak{c}

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

In

Z

we have

(n)\cap(m)=\operatorname{lcm}(n,m)Z

since

(n)\cap(m)

is the set of integers that are divisible by both

n

and .

Let

R=C[x,y,z,w]

and let . Then,

ak{a}+ak{b}=(z,w,x+z,y+w)=(x,y,z,w)

and

ak{a}+ak{c}=(z,w,x)

ak{a}ak{b}=(z(x+z),z(y+w),w(x+z),w(y+w))=(z2+xz,zy+wz,wx+wz,wy+w2)

ak{a}ak{c}=(xz+z2,zw,xw+zw,w2)

ak{a}\capak{b}=ak{a}ak{b}

while

ak{a}\capak{c}=(w,xz+z2) ≠ ak{a}ak{c}

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

See main article: 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.

J=\operatorname{Jac}(R)

of R is the intersection of all primitive ideals. Equivalently,

J=capak{mmaximalideals

} \mathfrak.Indeed, if

M

is a simple module and x is a nonzero element in M, then

Rx=M

and

R/\operatorname{Ann}(M)=R/\operatorname{Ann}(x)\simeqM

, meaning

\operatorname{Ann}(M)

is a maximal ideal. Conversely, if

ak{m}

is a maximal ideal, then

ak{m}

is the annihilator of the simple R-module . There is also another characterization (the proof is not hard):

J=\{x\inR\mid1-yxisaunitelementforeveryy\inR\}.

For a not-necessarily-commutative ring, it is a general fact that

1-yx

is a unit element if and only if

1-xy

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,

J ⋅ (M/L)=0

and so, a contradiction. Since a nonzero finitely generated module admits a maximal submodule, in particular, one has:

If

JM=M

and M is finitely generated, then .

A maximal ideal is a prime ideal and so one has

\operatorname{nil}(R)= capak{pprimeideals

} \mathfrak \subset \operatorname(R)where the intersection on the left is called the nilradical of R. As it turns out,

\operatorname{nil}(R)

is also the set of nilpotent elements of R.

If R is an Artinian ring, then

\operatorname{Jac}(R)

is nilpotent and . (Proof: first note the DCC implies

Jn=Jn+1

for some n. If (DCC)

ak{a}\supsetneq\operatorname{Ann}(Jn)

is an ideal properly minimal over the latter, then

J ⋅ (ak{a}/\operatorname{Ann}(Jn))=0

. That is,, a contradiction.)

Extension and contraction of an ideal

Let A and B be two commutative rings, and let f : A → B be a ring homomorphism. If

ak{a}

is an ideal in A, then

f(ak{a})

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

ak{a}e

of

ak{a}

in B is defined to be the ideal in B generated by . Explicitly,

ak{a}e=\{\sumyif(xi):xi\inak{a},yi\inB\}

If

ak{b}

is an ideal of B, then

f-1(ak{b})

is always an ideal of A, called the contraction

ak{b}c

of

ak{b}

to A.

Assuming f : A → B is a ring homomorphism,

ak{a}

is an ideal in A,

ak{b}

is an ideal in B, then:

ak{b}

is prime in B

⇒

ak{b}c

is prime in A.

ak{a}ec\supseteqak{a}

ak{b}ce\subseteqak{b}

It is false, in general, that

ak{a}

being prime (or maximal) in A implies that

ak{a}e

is prime (or maximal) in B. Many classic examples of this stem from algebraic number theory. For example, embedding . In

B=Z\left\lbracki\right\rbrack

, the element 2 factors as

2=(1+i)(1-i)

where (one can show) neither of

1+i,1-i

are units in B. So

(2)e

is not prime in B (and therefore not maximal, as well). Indeed,

(1\pmi)2=\pm2i

shows that,, and therefore .

On the other hand, if f is surjective and

ak{a}\supseteq\kerf

then:

ak{a}ec=ak{a}

and .

ak{a}

is a prime ideal in A

\Leftrightarrow

ak{a}e

is a prime ideal in B.

ak{a}

is a maximal ideal in A

\Leftrightarrow

ak{a}e

is a maximal ideal in B.

ak{a}=ak{p}

of A under extension is one of the central problems of algebraic number theory.

The following is sometimes useful: a prime ideal

ak{p}

is a contraction of a prime ideal if and only if . (Proof: Assuming the latter, note

ak{p}eBak{p

} = B_ \Rightarrow \mathfrak^e intersects, a contradiction. Now, the prime ideals of

Bak{p

} correspond to those in B that are disjoint from . Hence, there is a prime ideal

ak{q}

of B, disjoint from, such that

ak{q}Bak{p

} is a maximal ideal containing . One then checks that

ak{q}

lies over . The converse is obvious.)

Generalizations

Ideals can be generalized to any monoid object, where

R

is the object where the monoid structure has been forgotten. A left ideal of

R

is a subobject

I

that "absorbs multiplication from the left by elements of "; that is,

I

is a left ideal if it satisfies the following two conditions:

I

is a subobject of

R

  1. For every

r\in(R, ⊗ )

and every, the product

r ⊗ x

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

R

is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.

An ideal can also be thought of as a specific type of -module. If we consider

R

as a left

R

-module (by left multiplication), then a left ideal

I

is really just a left sub-module of . In other words,

I

is a left (right) ideal of

R

if and only if it is a left (right)

R

-module that is a subset of .

I

is a two-sided ideal if it is a sub-

R

-bimodule of .

Example: If we let, an ideal of

Z

is an abelian group that is a subset of, i.e.

mZ

for some . So these give all the ideals of .

See also

References

External links

Notes and References

  1. Book: John Stillwell. Mathematics and its history. 2010. 439.
  2. Book: Harold M. Edwards. Fermat's last theorem. A genetic introduction to algebraic number theory. 1977. 76.
  3. Book: Everest G., Ward T.. An introduction to number theory. 2005. 83.
  4. Some authors call the zero and unit ideals of a ring R the trivial ideals of R.
  5. 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.
  6. Because simple commutative rings are fields. See Book: Lam. 2001. [{{Google books|plainurl=y|id=f15FyZuZ3-4C|page=39|text=simple commutative rings}} A First Course in Noncommutative Rings]. 39.
  7. Book: Matsumura, Hideyuki . Hideyuki Matsumura . 1987 . Commutative Ring Theory . Cambridge . Cambridge University Press . 132 . 9781139171762.
  8. Web site: ideals. www.math.uiuc.edu. 2017-01-14. https://web.archive.org/web/20170116190119/http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_ideals.html. 2017-01-16. dead.
  9. Web site: sums, products, and powers of ideals. www.math.uiuc.edu. 2017-01-14. https://web.archive.org/web/20170116185903/http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_sums_cm_spproducts_cm_spand_sppowers_spof_spideals.html. 2017-01-16. dead.
  10. Web site: intersection of ideals. www.math.uiuc.edu. 2017-01-14. https://web.archive.org/web/20170116185829/http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_intersection_spof_spideals.html. 2017-01-16. dead.

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Ideal (ring theory)".

Except where otherwise indicated, Everything.Explained.Today is © Copyright 2009-2024, A B Cryer, All Rights Reserved. Cookie policy.