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Ideal (ring theory)

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Submodule of a mathematical ring

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Inmathematics, and more specifically inring theory, anideal of aring is a specialsubset of its elements. Ideals generalize certain subsets of theintegers, such as theeven 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; theseclosure andabsorption properties are the defining properties of an ideal. An ideal can be used to construct aquotient ring in a way similar to how, ingroup theory, anormal subgroup can be used to construct aquotient group.

Among the integers, the ideals correspond one-for-one with thenon-negative integers: in this ring, every ideal is aprincipal 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, theprime ideals of a ring are analogous toprime numbers, and theChinese remainder theorem can be generalized to ideals. There is a version ofunique prime factorization for the ideals of aDedekind domain (a type of ring important innumber theory).

The related, but distinct, concept of anideal inorder theory is derived from the notion of an ideal in ring theory. Afractional ideal is a generalization of an ideal, and the usual ideals are sometimes calledintegral ideals for clarity.

History

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Ernst Kummer invented the concept ofideal 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 ofDirichlet's bookVorlesungen über Zahlentheorie, to which Dedekind had added many supplements.^[1]^[2]^[3]Later the notion was extended beyond number rings to the setting of polynomial rings and other commutative rings byDavid Hilbert and especiallyEmmy Noether.

Definitions

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Given aring $R {\displaystyle R}$ , aleft ideal is a subset $I {\displaystyle I}$ of $R {\displaystyle R}$ that is asubgroup of theadditive group of $R {\displaystyle R}$ that isclosed under left multiplication by elements of⁠ $R {\displaystyle R}$ ⁠; that is, for every $r\in R$ and every⁠ $x,y\in I$ ⁠, one has:^[4]

⁠ $x+y\in I$ ⁠
⁠ $-x\in I$ ⁠
⁠ $rx\in I$ ⁠

In other words, a left ideal is a leftsubmodule of $R {\displaystyle R}$ , considered as aleft module over itself.^[5]

Aright ideal is defined similarly, with the condition $rx\in I$ replaced by⁠ $xr\in I$ ⁠. Atwo-sided ideal is a left ideal that is also a right ideal.

If the ring iscommutative, the definitions of left, right, and two-sided ideal coincide, and one talks simply of anideal. In the non-commutative case, "ideal" is often used instead of "two-sided ideal".

Since an ideal $I {\displaystyle I}$ is anabelian subgroup, the relation between⁠ $x {\displaystyle x}$ ⁠ and⁠ $y {\displaystyle y}$ ⁠ defined by

x-y\in I

is anequivalence relation on $R {\displaystyle R}$ , and the set ofequivalence classes is an abelian group denoted⁠ $R/I$ ⁠ and called thequotient of $R {\displaystyle R}$ by $I {\displaystyle I}$ .^[6] If $I {\displaystyle I}$ is a left or a right ideal, the quotient⁠ $R/I$ ⁠ is a left or right⁠ $R {\displaystyle R}$ ⁠-module, respectively.

If the ideal $I {\displaystyle I}$ is two-sided, the quotient $R/I$ is a ring,^[7] and the function

R\to R/I

that associates to each element of $R {\displaystyle R}$ its equivalence class is asurjective ring homomorphism that has the ideal as itskernel.^[8] 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

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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 arng. For a rng $R {\displaystyle R}$ , aleft ideal $I {\displaystyle I}$ is a subrng with the additional property that $r x {\displaystyle rx}$ is in $I {\displaystyle I}$ for every $r\in R$ and every $x\in I$ . (Right and two-sided ideals are defined similarly.) For a ring, an ideal $I {\displaystyle I}$ (say a left ideal) is rarely a subring; since a subring shares the same multiplicative identity with the ambient ring $R {\displaystyle R}$ , if $I {\displaystyle I}$ were a subring, for every $r\in R$ , we have $r=r1\in I;$ i.e., $I=R$ .

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

Traditionally, ideals are denoted usingFraktur lower-case letters, generally the first few letters ( ${\mathfrak {a}},{\mathfrak {b}},{\mathfrak {c}}$ , etc.) for generic ideals, ${\mathfrak {m}}$ for maximal ideals, and ${\mathfrak {p}}$ (and sometimes ${\mathfrak {q}}$ ) for prime ideals. In modern texts, capital letters, like $I {\displaystyle I}$ or $J {\displaystyle J}$ (or $M {\displaystyle M}$ and $P {\displaystyle P}$ for maximal and prime ideals, respectively) are also commonly used.

Examples and properties

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(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 ringR, the setR itself forms a two-sided ideal ofR called theunit ideal. It is often also denoted by $(1) {\displaystyle (1)}$ since it is precisely the two-sided ideal generated (see below) by the unity⁠ $1_{R}$ ⁠. Also, the set $\{0_{R}\}$ consisting of only the additive identity 0_R forms a two-sided ideal called thezero ideal and is denoted by⁠ $(0) {\displaystyle (0)}$ ⁠.^{[note 1]} Every (left, right or two-sided) ideal contains the zero ideal and is contained in the unit ideal.^[9]
An (left, right or two-sided) ideal that is not the unit ideal is called aproper ideal (as it is aproper subset).^[10] Note: a left ideal ${\mathfrak {a}}$ is proper if and only if it does not contain a unit element, since if $u\in {\mathfrak {a}}$ is a unit element, then $r=(ru^{-1})u\in {\mathfrak {a}}$ for every⁠ $r\in R$ ⁠. Typically there are plenty of proper ideals. In fact, ifR is askew-field, then $(0),(1)$ are its only ideals and conversely: that is, a nonzero ringR is a skew-field if $(0),(1)$ are the only left (or right) ideals. (Proof: if $x {\displaystyle x}$ is a nonzero element, then the principal left ideal $R x {\displaystyle Rx}$ (see below) is nonzero and thus $Rx=(1)$ ; i.e., $yx=1$ for some nonzero⁠ $y {\displaystyle y}$ ⁠. Likewise, $zy=1$ for some nonzero $z {\displaystyle z}$ . Then $z=z(yx)=(zy)x=x$ .)
The evenintegers form an ideal in the ring $\mathbb {Z}$ 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 $2\mathbb {Z}$ ⁠. More generally, the set of all integers divisible by a fixed integer $n {\displaystyle n}$ is an ideal denoted⁠ $n\mathbb {Z}$ ⁠. In fact, every non-zero ideal of the ring $\mathbb {Z}$ is generated by its smallest positive element, as a consequence ofEuclidean division, so $\mathbb {Z}$ is aprincipal ideal domain.^[9]
The set of allpolynomials with real coefficients that are divisible by the polynomial $x^{2}+1$ is an ideal in the ring of all real-coefficient polynomials⁠ $\mathbb {R} [x]$ ⁠.
Take a ring $R {\displaystyle R}$ and positive integer⁠ $n {\displaystyle n}$ ⁠. For each⁠ $1\leq i\leq n$ ⁠, the set of all $n\times n$ matrices with entries in $R {\displaystyle R}$ whose $i {\displaystyle i}$ -th row is zero is a right ideal in the ring $M_{n}(R)$ of all $n\times n$ matrices with entries in⁠ $R {\displaystyle R}$ ⁠. It is not a left ideal. Similarly, for each⁠ $1\leq j\leq n$ ⁠, the set of all $n\times n$ matrices whose $j {\displaystyle j}$ -thcolumn is zero is a left ideal but not a right ideal.
The ring $C(\mathbb {R} )$ of allcontinuous functions $f {\displaystyle f}$ from $\mathbb {R}$ to $\mathbb {R}$ underpointwise multiplication contains the ideal of all continuous functions $f {\displaystyle f}$ such that⁠ $f(1)=0$ ⁠.^[11] Another ideal in $C(\mathbb {R} )$ is given by those functions that vanish for large enough arguments, i.e. those continuous functions $f {\displaystyle f}$ for which there exists a number $L>0$ such that $f(x)=0$ whenever⁠ $\vert x\vert >L$ ⁠.
A ring is called asimple ring if it is nonzero and has no two-sided ideals other than⁠ $(0),(1)$ ⁠. Thus, a skew-field is simple and a simple commutative ring is a field. Thematrix ring over a skew-field is a simple ring.
If $f:R\to S$ is aring homomorphism, then the kernel $\ker(f)=f^{-1}(0_{S})$ is a two-sided ideal of⁠ $R {\displaystyle R}$ ⁠.^[9] By definition,⁠ $f(1_{R})=1_{S}$ ⁠, and thus if $S {\displaystyle S}$ is not thezero ring (so⁠ $1_{S}\neq 0_{S}$ ⁠), then $\ker(f)$ is a proper ideal. More generally, for each left idealI ofS, the pre-image $f^{-1}(I)$ is a left ideal. IfI is a left ideal ofR, then $f(I)$ is a left ideal of the subring $f(R)$ ofS: unlessf is surjective, $f(I)$ need not be an ideal ofS; see also§ Extension and contraction of an ideal.
Ideal correspondence: Given a surjective ring homomorphism⁠ $f:R\to S$ ⁠, there is a bijective order-preserving correspondence between the left (resp. right, two-sided) ideals of $R {\displaystyle R}$ containing the kernel of $f {\displaystyle f}$ and the left (resp. right, two-sided) ideals of $S {\displaystyle S}$ : the correspondence is given by $I\mapsto f(I)$ and the pre-image⁠ $J\mapsto f^{-1}(J)$ ⁠. Moreover, for commutative rings, this bijective correspondence restricts to prime ideals, maximal ideals, and radical ideals (see theTypes of ideals section for the definitions of these ideals).
IfM is a leftR-module and $S\subset M$ a subset, then theannihilator $\operatorname {Ann} _{R}(S)=\{r\in R\mid rs=0,s\in S\}$ ofS is a left ideal. Given ideals ${\mathfrak {a}},{\mathfrak {b}}$ of a commutative ringR, theR-annihilator of $({\mathfrak {b}}+{\mathfrak {a}})/{\mathfrak {a}}$ is an ideal ofR called theideal quotient of ${\mathfrak {a}}$ by ${\mathfrak {b}}$ and is denoted by⁠ $({\mathfrak {a}}:{\mathfrak {b}})$ ⁠; it is an instance ofidealizer in commutative algebra.
Let ${\mathfrak {a}}_{i},i\in S$ be anascending chain of left ideals in a ringR; i.e., $S {\displaystyle S}$ is a totally ordered set and ${\mathfrak {a}}_{i}\subset {\mathfrak {a}}_{j}$ for each⁠ $i<j$ ⁠. Then the union $\textstyle \bigcup _{i\in S}{\mathfrak {a}}_{i}$ is a left ideal ofR. (Note: this fact remains true even ifR is without the unity 1.)
The above fact together withZorn's lemma proves the following: if $E\subset R$ is a possibly empty subset and ${\mathfrak {a}}_{0}\subset R$ is a left ideal that is disjoint fromE, then there is an ideal that is maximal among the ideals containing ${\mathfrak {a}}_{0}$ and disjoint fromE. (Again this is still valid if the ringR lacks the unity 1.) When $R\neq 0$ , taking ${\mathfrak {a}}_{0}=(0)$ and⁠ $E=\{1\}$ ⁠, in particular, there exists a left ideal that is maximal among proper left ideals (often simply called a maximal left ideal); seeKrull's theorem for more.
A left (resp. right, two-sided) ideal generated by a single elementx is called the principal left (resp. right, two-sided) ideal generated byx and is denoted by $R x {\displaystyle Rx}$ (resp.⁠ $x R, R x R {\displaystyle xR,RxR}$ ⁠). The principal two-sided ideal $R x R {\displaystyle RxR}$ is often also denoted by⁠ $(x)$ ⁠ or $\langle x\rangle$ .
An arbitrary union of ideals need not be an ideal, but the following is still true: given a possibly empty subsetX ofR, there is the smallest left ideal containingX, called the left ideal generated byX and is denoted by⁠ $R X {\displaystyle RX}$ ⁠. Such an ideal exists since it is the intersection of all left ideals containingX. Equivalently, $R X {\displaystyle RX}$ is the set of all the(finite) leftR-linear combinations of elements ofX overR:
$RX=\{r_{1}x_{1}+\dots +r_{n}x_{n}\mid n\in \mathbb {N} ,r_{i}\in R,x_{i}\in X\}$ (since such a span is the smallest left ideal containingX.)^{[note 2]} A right (resp. two-sided) ideal generated byX is defined in the similar way. For "two-sided", one has to use linear combinations from both sides; i.e.,
$RXR=\{r_{1}x_{1}s_{1}+\dots +r_{n}x_{n}s_{n}\mid n\in \mathbb {N} ,r_{i}\in R,s_{i}\in R,x_{i}\in X\}.$ If $X=\{x_{1},\dots ,x_{n}\}$ is afinite set, then $R X R {\displaystyle RXR}$ is also written as⁠ $(x_{1},\dots ,x_{n})$ ⁠ or $\langle x_{1},...,x_{n}\rangle$ . More generally, the two-sided ideal generated by a (finite or infinite) set of indexed ring elements $X=\{x_{i}\}_{i\in I}$ is denoted $(X)=(x_{i})_{i\in I}$ or $\langle X\rangle =\langle x_{i}\rangle _{i\in I}$ .
There is a bijective correspondence between ideals andcongruence relations (equivalence relations that respect the ring structure) on the ring: Given an ideal $I {\displaystyle I}$ of a ring⁠ $R {\displaystyle R}$ ⁠, let $x\sim y$ if⁠ $x-y\in I$ ⁠. Then $\sim$ is a congruence relation on⁠ $R {\displaystyle R}$ ⁠. Conversely, given a congruence relation $\sim$ on⁠ $R {\displaystyle R}$ ⁠, let⁠ $I=\{x\in R:x\sim 0\}$ ⁠. Then $I {\displaystyle I}$ is an ideal of⁠ $R {\displaystyle R}$ ⁠.

Types of ideals

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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 definefactor rings. Different types of ideals are studied because they can be used to construct different types of factor rings.

Maximal ideal: A proper idealI is called amaximal ideal if there exists no other proper idealJ withI a proper subset ofJ. The factor ring of a maximal ideal is asimple ring in general and is afield for commutative rings.^[12]
Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal.
Zero ideal: the ideal $\{0\}$ .^[13]
Unit ideal: the whole ring (being the ideal generated by $1 {\displaystyle 1}$ ).^[9]
Prime ideal: A proper ideal $I {\displaystyle I}$ is called aprime ideal if for any $a {\displaystyle a}$ and $b {\displaystyle b}$ in⁠ $R {\displaystyle R}$ ⁠, if $a b {\displaystyle ab}$ is in⁠ $I {\displaystyle I}$ ⁠, then at least one of $a {\displaystyle a}$ and $b {\displaystyle b}$ is in⁠ $I {\displaystyle I}$ ⁠. The factor ring of a prime ideal is aprime ring in general and is anintegral domain for commutative rings.^[14]
Radical ideal orsemiprime ideal: A proper idealI is calledradical orsemiprime if for anya in $R {\displaystyle R}$ , ifaⁿ is inI for somen, thena is inI. The factor ring of a radical ideal is asemiprime ring for general rings, and is areduced ring for commutative rings.
Primary ideal: An idealI is called aprimary ideal if for alla andb inR, ifab is inI, then at least one ofa andbⁿ is inI for somenatural numbern. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime.
Principal ideal: An ideal generated byone element.^[15]
Finitely generated ideal: This type of ideal isfinitely generated as a module.
Primitive ideal: A left primitive ideal is theannihilator of asimple leftmodule.
Irreducible ideal: An ideal is said to be irreducible if it cannot be written as an intersection of ideals that properly contain it.
Comaximal ideals: Two idealsI,J are said to becomaximal if $x+y=1$ for some $x\in I$ and⁠ $y\in J$ ⁠.
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 asystem of parameters.
Perfect ideal: A proper idealI in a Noetherian ring $R {\displaystyle R}$ is called aperfect ideal if itsgrade equals theprojective dimension of the associated quotient ring,^[16]⁠ ${\textrm {grade}}(I)={\textrm {proj}}\dim(R/I)$ ⁠. A perfect ideal isunmixed.
Unmixed ideal: A proper idealI in a Noetherian ring $R {\displaystyle R}$ is called anunmixed ideal (in height) if the height ofI is equal to the height of everyassociated primeP of $R/I$ . (This is stronger than saying that $R/I$ isequidimensional. See alsoequidimensional ring.

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 $R {\displaystyle R}$ is a commutative domain withquotient field $K {\displaystyle K}$ . Despite their names, fractional ideals are not necessarily ideals. A fractional ideal of $R {\displaystyle R}$ is an $R {\displaystyle R}$ -submodule⁠ $I {\displaystyle I}$ ⁠ of $K {\displaystyle K}$ for which there exists a non-zero $r\in R$ such that $rI\subseteq R$ . If the fractional ideal is contained entirely in $R {\displaystyle R}$ , then it is truly an ideal of $R {\displaystyle R}$ .
Invertible ideal: Usually an invertible idealA is defined as a fractional ideal for which there is another fractional idealB such thatAB =BA =R. Some authors may also apply "invertible ideal" to ordinary ring idealsA andB withAB =BA =R in rings other than domains.

Ideal operations

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The sum and product of ideals are defined as follows. For ${\mathfrak {a}}$ and⁠ ${\mathfrak {b}}$ ⁠, left (resp. right) ideals of a ringR, their sum is

{\mathfrak {a}}+{\mathfrak {b}}:=\{a+b\mid a\in {\mathfrak {a}}{\mbox{ and }}b\in {\mathfrak {b}}\}

which is a left (resp. right) ideal,and, if ${\mathfrak {a}},{\mathfrak {b}}$ are two-sided,

{\mathfrak {a}}{\mathfrak {b}}:=\{a_{1}b_{1}+\dots +a_{n}b_{n}\mid a_{i}\in {\mathfrak {a}}{\mbox{ and }}b_{i}\in {\mathfrak {b}},i=1,2,\dots ,n;{\mbox{ for }}n=1,2,\dots \},

i.e. the product is the ideal generated by all products of the formab witha in ${\mathfrak {a}}$ andb in⁠ ${\mathfrak {b}}$ ⁠.

Note ${\mathfrak {a}}+{\mathfrak {b}}$ is the smallest left (resp. right) ideal containing both ${\mathfrak {a}}$ and ${\mathfrak {b}}$ (or the union⁠ ${\mathfrak {a}}\cup {\mathfrak {b}}$ ⁠), while the product ${\mathfrak {a}}{\mathfrak {b}}$ is contained in the intersection of ${\mathfrak {a}}$ and⁠ ${\mathfrak {b}}$ ⁠.

The distributive law holds for two-sided ideals⁠ ${\mathfrak {a}},{\mathfrak {b}},{\mathfrak {c}}$ ⁠,

⁠ ${\mathfrak {a}}({\mathfrak {b}}+{\mathfrak {c}})={\mathfrak {a}}{\mathfrak {b}}+{\mathfrak {a}}{\mathfrak {c}}$ ⁠,
⁠ $({\mathfrak {a}}+{\mathfrak {b}}){\mathfrak {c}}={\mathfrak {a}}{\mathfrak {c}}+{\mathfrak {b}}{\mathfrak {c}}$ ⁠.

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

{\mathfrak {a}}\cap ({\mathfrak {b}}+{\mathfrak {c}})\supset {\mathfrak {a}}\cap {\mathfrak {b}}+{\mathfrak {a}}\cap {\mathfrak {c}}

where the equality holds if ${\mathfrak {a}}$ contains ${\mathfrak {b}}$ or ${\mathfrak {c}}$ .

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 acomplete modular lattice. The lattice is not, in general, adistributive lattice. The three operations of intersection, sum (or join), and product make the set of ideals of a commutative ring into aquantale.

If ${\mathfrak {a}},{\mathfrak {b}}$ are ideals of a commutative ringR, then ${\mathfrak {a}}\cap {\mathfrak {b}}={\mathfrak {a}}{\mathfrak {b}}$ in the following two cases (at least)

${\mathfrak {a}}+{\mathfrak {b}}=(1)$
${\mathfrak {a}}$ is generated by elements that form aregular sequence modulo⁠ ${\mathfrak {b}}$ ⁠.

(More generally, the difference between a product and an intersection of ideals is measured by theTor functor:⁠ $\operatorname {Tor} _{1}^{R}(R/{\mathfrak {a}},R/{\mathfrak {b}})=({\mathfrak {a}}\cap {\mathfrak {b}})/{\mathfrak {a}}{\mathfrak {b}}$ ⁠.^[17])

An integral domain is called aDedekind domain if for each pair of ideals ${\mathfrak {a}}\subset {\mathfrak {b}}$ , there is an ideal ${\mathfrak {c}}$ such that⁠ ${\mathfrak {\mathfrak {a}}}={\mathfrak {b}}{\mathfrak {c}}$ ⁠.^[18] 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

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In $\mathbb {Z}$ we have

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

since $(n)\cap (m)$ is the set of integers that are divisible by both $n {\displaystyle n}$ and⁠ $m {\displaystyle m}$ ⁠.

Let $R=\mathbb {C} [x,y,z,w]$ and let⁠ ${\mathfrak {a}}=(z,w),{\mathfrak {b}}=(x+z,y+w),{\mathfrak {c}}=(x+z,w)$ ⁠. Then,

${\mathfrak {a}}+{\mathfrak {b}}=(z,w,x+z,y+w)=(x,y,z,w)$ and ${\mathfrak {a}}+{\mathfrak {c}}=(z,w,x)$
${\mathfrak {a}}{\mathfrak {b}}=(z(x+z),z(y+w),w(x+z),w(y+w))=(z^{2}+xz,zy+wz,wx+wz,wy+w^{2})$
${\mathfrak {a}}{\mathfrak {c}}=(xz+z^{2},zw,xw+zw,w^{2})$
${\mathfrak {a}}\cap {\mathfrak {b}}={\mathfrak {a}}{\mathfrak {b}}$ while ${\mathfrak {a}}\cap {\mathfrak {c}}=(w,xz+z^{2})\neq {\mathfrak {a}}{\mathfrak {c}}$

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 usingMacaulay2.^[19]^[20]^[21]

Radical of a ring

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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.

LetR be a commutative ring. By definition, aprimitive ideal ofR is the annihilator of a (nonzero)simpleR-module. TheJacobson radical $J=\operatorname {Jac} (R)$ ofR is the intersection of all primitive ideals. Equivalently,

J=\bigcap _{{\mathfrak {m}}{\text{ maximal ideals}}}{\mathfrak {m}}.

Indeed, if $M {\displaystyle M}$ is a simple module andx is a nonzero element inM, then $Rx=M$ and $R/\operatorname {Ann} (M)=R/\operatorname {Ann} (x)\simeq M$ , meaning $\operatorname {Ann} (M)$ is a maximal ideal. Conversely, if ${\mathfrak {m}}$ is a maximal ideal, then ${\mathfrak {m}}$ is the annihilator of the simpleR-module⁠ $R/{\mathfrak {m}}$ ⁠. There is also another characterization (the proof is not hard):

J=\{x\in R\mid 1-yx\,{\text{ is a unit element for every }}y\in R\}.

For a not-necessarily-commutative ring, it is a general fact that $1-yx$ is aunit 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: ifM is a module such that⁠ $JM=M$ ⁠, thenM does not admit amaximal submodule, since if there is a maximal submodule⁠ $L\subsetneq M$ ⁠, $J\cdot (M/L)=0$ and so⁠ $M=JM\subset L\subsetneq M$ ⁠, a contradiction. Since a nonzerofinitely generated module admits a maximal submodule, in particular, one has:

JM=M

andM is finitely generated, then⁠

M=0

⁠.

A maximal ideal is a prime ideal and so one has

\operatorname {nil} (R)=\bigcap _{{\mathfrak {p}}{\text{ prime ideals }}}{\mathfrak {p}}\subset \operatorname {Jac} (R)

where the intersection on the left is called thenilradical ofR. As it turns out, $\operatorname {nil} (R)$ is also the set ofnilpotent elements ofR.

IfR is anArtinian ring, then $\operatorname {Jac} (R)$ is nilpotent and⁠ $\operatorname {nil} (R)=\operatorname {Jac} (R)$ ⁠. (Proof: first note the DCC implies $J^{n}=J^{n+1}$ for somen. If (DCC) ${\mathfrak {a}}\supsetneq \operatorname {Ann} (J^{n})$ is an ideal properly minimal over the latter, then $J\cdot ({\mathfrak {a}}/\operatorname {Ann} (J^{n}))=0$ . That is,⁠ $J^{n}{\mathfrak {a}}=J^{n+1}{\mathfrak {a}}=0$ ⁠, a contradiction.)

Extension and contraction of an ideal

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LetA andB be twocommutative rings, and let $f:A\to B$ be aring homomorphism. If ${\mathfrak {a}}$ is an ideal inA, then $f({\mathfrak {a}})$ need not be an ideal inB (e.g. takef to be theinclusion of the ring of integers $\mathbb {Z}$ into the field of rationals $\mathbb {Q}$ ). Theextension ${\mathfrak {a}}^{e}$ of ${\mathfrak {a}}$ inB is defined to be the ideal inB generated by⁠ $f({\mathfrak {a}})$ ⁠. Explicitly,

{\mathfrak {a}}^{e}=f({\mathfrak {a}})B={\Big \{}\sum y_{i}f(x_{i}):x_{i}\in {\mathfrak {a}},y_{i}\in B{\Big \}}.

By abuse of notation, ${\mathfrak {a}}B$ is another common notation for this ideal extension.

If ${\mathfrak {b}}$ is an ideal ofB, then $f^{-1}({\mathfrak {b}})$ is always an ideal ofA, called thecontraction ${\mathfrak {b}}^{c}$ of ${\mathfrak {b}}$ toA.

Assuming $f:A\to B$ is a ring homomorphism, ${\mathfrak {a}}$ is an ideal inA, ${\mathfrak {b}}$ is an ideal inB, then:

${\mathfrak {b}}$ is prime inB $\Rightarrow$ ${\mathfrak {b}}^{c}$ is prime inA,
${\mathfrak {a}}^{ec}\supseteq {\mathfrak {a}},$
${\mathfrak {b}}^{ce}\subseteq {\mathfrak {b}}.$

It is false, in general, that ${\mathfrak {a}}$ being prime (or maximal) inA implies that ${\mathfrak {a}}^{e}$ is prime (or maximal) inB. Many classic examples of this stem from algebraic number theory. For example, consider theembedding $\mathbb {Z} \to \mathbb {Z} \left\lbrack i\right\rbrack .$ In $B=\mathbb {Z} \left\lbrack i\right\rbrack$ , the element 2 factors as $2=(1+i)(1-i)$ where (one can show) neither of $1+i,1-i$ are units inB. So $(2)^{e}$ is not prime inB (and therefore not maximal, as well). Indeed, $(1\pm i)^{2}=\pm 2i$ shows that⁠ $(1+i)=((1-i)-(1-i)^{2})$ ⁠,⁠ $(1-i)=((1+i)-(1+i)^{2})$ ⁠, and therefore⁠ $(2)^{e}=(1+i)^{2}$ ⁠.

On the other hand, iff issurjective and ${\mathfrak {a}}\supseteq \ker f$ then:

${\mathfrak {a}}^{ec}={\mathfrak {a}}$ and ${\mathfrak {b}}^{ce}={\mathfrak {b}},$
${\mathfrak {a}}$ is aprime ideal inA $\Leftrightarrow$ ${\mathfrak {a}}^{e}$ is a prime ideal inB,
${\mathfrak {a}}$ is amaximal ideal inA $\Leftrightarrow$ ${\mathfrak {a}}^{e}$ is a maximal ideal inB.

Remark: LetK be afield extension ofL, and letB andA be therings of integers ofK andL, respectively. ThenB is anintegral extension ofA, and we letf be theinclusion map fromA toB. The behaviour of aprime ideal ${\mathfrak {a}}={\mathfrak {p}}$ ofA under extension is one of the central problems ofalgebraic number theory.

The following is sometimes useful:^[22] a prime ideal ${\mathfrak {p}}$ is a contraction of a prime ideal if and only if ${\mathfrak {p}}={\mathfrak {p}}^{ec}$ . (Proof: Assuming the latter, note ${\mathfrak {p}}^{e}B_{\mathfrak {p}}=B_{\mathfrak {p}}\Rightarrow {\mathfrak {p}}^{e}$ intersects $A-{\mathfrak {p}}$ , a contradiction. Now, the prime ideals of $B_{\mathfrak {p}}$ correspond to those inB that are disjoint from $A-{\mathfrak {p}}$ . Hence, there is a prime ideal ${\mathfrak {q}}$ ofB, disjoint from $A-{\mathfrak {p}}$ , such that ${\mathfrak {q}}B_{\mathfrak {p}}$ is a maximal ideal containing ${\mathfrak {p}}^{e}B_{\mathfrak {p}}$ . One then checks that ${\mathfrak {q}}$ lies over ${\mathfrak {p}}$ . The converse is obvious.)

Generalizations

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Ideals can be generalized to anymonoid object⁠ $(R,\otimes )$ ⁠, where $R {\displaystyle R}$ is the object where themonoid structure has beenforgotten. Aleft ideal of $R {\displaystyle R}$ is asubobject $I {\displaystyle I}$ that "absorbs multiplication from the left by elements of⁠ $R {\displaystyle R}$ ⁠"; that is, $I {\displaystyle I}$ is aleft ideal if it satisfies the following two conditions:

$I {\displaystyle I}$ is asubobject of $R {\displaystyle R}$
For every $r\in (R,\otimes )$ and every⁠ $x\in (I,\otimes )$ ⁠, the product $r\otimes x$ is in⁠ $(I,\otimes )$ ⁠.

Aright ideal is defined with the condition "⁠ $r\otimes x\in (I,\otimes )$ ⁠" replaced by "'⁠ $x\otimes r\in (I,\otimes )$ ⁠". Atwo-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. When $R {\displaystyle R}$ is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the termideal is used alone.

Notes

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^Some authors call the zero and unit ideals of a ringR thetrivial ideals ofR.
^IfR does not have a unit, then the internal descriptions above must be modified slightly. In addition to the finite sums of products of things inX with things inR, we must allow the addition ofn-fold sums of the formx +x + ... +x, andn-fold sums of the form(−x) + (−x) + ... + (−x) for everyx inX and everyn in the natural numbers. WhenR has a unit, this extra requirement becomes superfluous.

References

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^^a ^bJohn Stillwell (2010).Mathematics and its history. p. 439.
^Harold M. Edwards (1977).Fermat's last theorem. A genetic introduction to algebraic number theory. p. 76.
^Everest G., Ward T. (2005).An introduction to number theory. p. 83.
^Dummit & Foote 2004, p. 242
^Dummit & Foote 2004, § 10.1., Examples (1).
^Dummit & Foote 2004, § 10.1., Proposition 3.
^Dummit & Foote 2004, Ch. 7, Proposition 6.
^Dummit & Foote 2004, Ch. 7, Theorem 7.
^^a ^b ^c ^dDummit & Foote (2004), p. 243.
^Lang (2005), Section III.2.
^Dummit & Foote (2004), p. 244.
^Because simple commutative rings are fields. SeeLam (2001).A First Course in Noncommutative Rings. p. 39.
^"Zero ideal".Math World. 22 Aug 2024.
^Dummit & Foote (2004), p. 255.
^Dummit & Foote (2004), p. 251.
^Matsumura, Hideyuki (1987).Commutative Ring Theory. Cambridge: Cambridge University Press. p. 132.ISBN 9781139171762.
^Eisenbud 1995, Exercise A 3.17
^Milnor (1971), p. 9.
^"ideals".www.math.uiuc.edu. Archived fromthe original on 2017-01-16. Retrieved2017-01-14.
^"sums, products, and powers of ideals".www.math.uiuc.edu. Archived fromthe original on 2017-01-16. Retrieved2017-01-14.
^"intersection of ideals".www.math.uiuc.edu. Archived fromthe original on 2017-01-16. Retrieved2017-01-14.
^Atiyah & Macdonald (1969), Proposition 3.16.