Ideal (ring theory)

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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.^[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.^[1]^[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 $R$ , a left ideal is a subset $I$ of $R$ that is a subgroup of the additive group of $R$ that is closed under left multiplication by elements of Template:Tmath; that is, $0\in I$ and for every $r\in R$ and every Template:Tmath, one has^[4]

In other words, a left ideal is a left submodule of $R$ , considered as a left module over itself.^[5]

A right ideal is defined similarly, with the condition $rx\in I$ replaced by Template:Tmath. A two-sided ideal 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".

Since an ideal $I$ is an abelian subgroup, the relation between Template:Tmath and Template:Tmath defined by

x-y\in I

is an equivalence relation on $R$ , and the set of equivalence classes is an abelian group denoted Template:Tmath and called the quotient of $R$ by $I$ .^[6] If $I$ is a left or a right ideal, the quotient Template:Tmath is a left or right Template:Tmath-module, respectively.

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

R\to R/I

that associates to each element of $R$ its equivalence class is a surjective ring homomorphism that has the ideal as its kernel.^[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

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". (See also Ring (mathematics) § History.) For a rng $R$ , a left ideal $I$ is a subrng with the additional property that $rx$ is in $I$ for every $r\in R$ and every $x\in I$ . (Right and two-sided ideals are defined similarly.) For a ring, an ideal $I$ (say a left ideal) is rarely a subring; since a subring shares the same multiplicative identity with the ambient ring $R$ , if $I$ 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 for non-associative rings.

For algebras, we additionally assume that an ideal is a linear subspace. If a $k$ -algebra $A$ is unital, then any ring-ideal $I\subseteq A$ is an algebra-ideal, since $ax+by=(a1)x+(b1)y\in I$ when $x,y\in I$ and $a,b\in k$ . For non-unital algebras, there can exist ring-ideals that are not algebra-ideals, for example if $xy=0$ for all $x,y\in A$ , then ring-ideals of $A$ are subrings of $A$ , while algebra-ideals of $A$ are linear subspaces of $A$ .

Traditionally, ideals are denoted using Fraktur lower-case letters, generally the first few letters ( ${\mathfrak {a}},{\mathfrak {b}},{\mathfrak {c}}$ , etc.) for generic ideals, ${\mathfrak {m}}$ for maximal ideals, and ${\mathfrak {p}}$ for prime ideals. In modern texts, capital letters, like $I$ or $J$ (or $M$ and $P$ 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. It is often also denoted by $(1)$ since it is precisely the two-sided ideal generated (see below) by the unity Template:Tmath. Also, the set $\{0_{R}\}$ consisting of only the additive identity 0_R forms a two-sided ideal called the zero ideal and is denoted by Template:Tmath.^{[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 a proper ideal (as it is a proper 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 Template:Tmath. 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 Template:Tmath. Likewise, $zy=1$ for some nonzero $z$ . Then $z=z(yx)=(zy)x=x$ .)
The even integers 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 Template:Tmath. More generally, the set of all integers divisible by a fixed integer $n$ is an ideal denoted Template:Tmath. In fact, every non-zero ideal of the ring $\mathbb {Z}$ is generated by its smallest positive element, as a consequence of Euclidean division, so $\mathbb {Z}$ is a principal ideal domain.^[9]
The set of all polynomials with real coefficients that are divisible by the polynomial $x^{2}+1$ is an ideal in the ring of all real-coefficient polynomials Template:Tmath.
Take a ring $R$ and positive integer Template:Tmath. For each Template:Tmath, the set of all $n\times n$ matrices with entries in $R$ whose $i$ -th row is zero is a right ideal in the ring $M_{n}(R)$ of all $n\times n$ matrices with entries in Template:Tmath. It is not a left ideal. Similarly, for each Template:Tmath, the set of all $n\times n$ matrices whose $j$ -th column is zero is a left ideal but not a right ideal.
The ring $C(\mathbb {R} )$ of all continuous functions $f$ from $\mathbb {R}$ to $\mathbb {R}$ under pointwise multiplication contains the ideal of all continuous functions $f$ such that Template:Tmath.^[11] Another ideal in $C(\mathbb {R} )$ 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 Template:Tmath.
A ring is called a simple ring if it is nonzero and has no two-sided ideals other than Template:Tmath. 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 $f:R\to S$ is a ring homomorphism, then the kernel $\ker(f)=f^{-1}(0_{S})$ is a two-sided ideal of Template:Tmath.^[9] By definition, Template:Tmath, and thus if $S$ is not the zero ring (so Template:Tmath), 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 § Extension and contraction of an ideal.
Ideal correspondence: Given a surjective ring homomorphism Template:Tmath, there is a bijective order-preserving correspondence between the left (resp. right, two-sided) ideals of $R$ containing the kernel of $f$ and the left (resp. right, two-sided) ideals of $S$ : the correspondence is given by $I\mapsto f(I)$ and the pre-image Template:Tmath. 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 and $S\subset M$ a subset, then the annihilator $\operatorname {Ann} _{R}(S)=\{r\in R\mid rs=0,s\in S\}$ of S is a left ideal. Given ideals ${\mathfrak {a}},{\mathfrak {b}}$ of a commutative ring R, the R-annihilator of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathfrak{b} + \mathfrak{a})/\mathfrak{a}} is an ideal of R called the ideal quotient of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{b}} and is denoted by Template:Tmath; it is an instance of idealizer in commutative algebra.
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}_i, i \in S} be an ascending chain of left ideals in a ring R; i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a totally ordered set and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}_i \subset \mathfrak{a}_j} for each Template:Tmath. Then the union Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle \bigcup_{i \in S} \mathfrak{a}_i} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E \subset R} is a possibly empty subset and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}_0 \subset R} is a left ideal that is disjoint from E, then there is an ideal that is maximal among the ideals containing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}_0} and disjoint from E. (Again this is still valid if the ring R lacks the unity 1.) When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R \ne 0} , taking Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}_0 = (0)} and Template:Tmath, 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 and is denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Rx} (resp. Template:Tmath). The principal two-sided ideal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle RxR} is often also denoted by Template:Tmath or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x \rangle} .
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 Template:Tmath. Such an ideal exists since it is the intersection of all left ideals containing X. Equivalently, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle RX} is the set of all the (finite) left R-linear combinations of elements of X over R:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle RX = \{r_1x_1+\dots+r_nx_n \mid n\in\mathbb{N}, r_i\in R, x_i\in X\}} (since such a span is the smallest left ideal containing X.)^{[note 2]} 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.,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle RXR = \{r_1x_1s_1+\dots+r_nx_ns_n \mid n\in\mathbb{N}, r_i\in R,s_i\in R, x_i\in X\} .} If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = \{ x_1, \dots, x_n \}} is a finite set, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle RXR} is also written as Template:Tmath or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x_1,...,x_n\rangle} . More generally, the two-sided ideal generated by a (finite or infinite) set of indexed ring elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=\{ x_i\}_{i\in I}} is denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X)=(x_i)_{i\in I}} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle X\rangle=\langle x_i\rangle_{i\in I}} .
There is a bijective correspondence between ideals and congruence relations (equivalence relations that respect the ring structure) on the ring: Given an ideal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} of a ring Template:Tmath, let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\sim y} if Template:Tmath. Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim} is a congruence relation on Template:Tmath. Conversely, given a congruence relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim} on Template:Tmath, let Template:Tmath. Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is an ideal of Template:Tmath.

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.^[12]
Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal.
Zero ideal: the ideal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{0\}} .^[13]
Unit ideal: the whole ring (being the ideal generated by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} ).^[9]
Prime ideal: A proper ideal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is called a prime ideal if for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} in Template:Tmath, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ab} is in Template:Tmath, then at least one of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} is in Template:Tmath. The factor ring of a prime ideal is a prime ring in general and is an integral domain for commutative rings.^[14]
Radical ideal or semiprime ideal: A proper ideal $I$ is called radical or semiprime if for any $a$ in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.^[15]
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 that properly contain it.
Comaximal ideals: Two ideals $I$ , $J$ are said to be comaximal if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x + y = 1} for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in I} and Template:Tmath.
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.
Perfect ideal: A proper ideal $I$ in a Noetherian ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is called a perfect ideal if its grade equals the projective dimension of the associated quotient ring,^[16] Template:Tmath. A perfect ideal is unmixed.
Unmixed ideal: A proper ideal $I$ in a Noetherian ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is called an unmixed ideal (in height) if the height of $I$ is equal to the height of every associated prime $P$ of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R/I} . (This is stronger than saying that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Fractional ideal: This is usually defined when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a commutative domain with quotient field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} . Despite their names, fractional ideals are not necessarily ideals. A fractional ideal of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} -submodule Template:Tmath of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} for which there exists a non-zero Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \in R} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle rI \subseteq R} . If the fractional ideal is contained entirely in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} , then it is truly an ideal of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}} and Template:Tmath, left (resp. right) ideals of a ring R, their sum is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}, \mathfrak{b}} are two-sided,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a} \mathfrak{b}:=\{a_1b_1+ \dots + a_nb_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 form ab with a in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}} and b in Template:Tmath.

Note Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a} + \mathfrak{b}} is the smallest left (resp. right) ideal containing both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{b}} (or the union Template:Tmath), while the product Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}\mathfrak{b}} is contained in the intersection of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}} and Template:Tmath.

The distributive law holds for two-sided ideals Template:Tmath,

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a} \cap (\mathfrak{b} + \mathfrak{c}) \supset \mathfrak{a} \cap \mathfrak{b} + \mathfrak{a} \cap \mathfrak{c}}

where the equality holds if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}} contains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{b}} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}, \mathfrak{b}} are ideals of a commutative ring R, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a} \cap \mathfrak{b} = \mathfrak{a} \mathfrak{b}} in the following two cases (at least)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a} + \mathfrak{b} = (1)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}} is generated by elements that form a regular sequence modulo Template:Tmath.

(More generally, the difference between a product and an intersection of ideals is measured by the Tor functor: Template:Tmath.^[17])

An integral domain is called a Dedekind domain if for each pair of ideals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a} \subset \mathfrak{b}} , there is an ideal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{c}} such that Template:Tmath.^[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 the fundamental theorem of arithmetic.

Examples of ideal operations

In Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}} we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n)\cap(m) = \operatorname{lcm}(n,m)\mathbb{Z}}

since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n)\cap(m)} is the set of integers that are divisible by both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} and Template:Tmath.

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = \mathbb{C}[x,y,z,w]} and let Template:Tmath. Then,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a} + \mathfrak{b} = (z,w, x+z, y+w) = (x, y, z, w)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a} + \mathfrak{c} = (z, w, x)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}\mathfrak{c} = (xz + z^2, zw, xw + zw, w^2)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a} \cap \mathfrak{b} = \mathfrak{a}\mathfrak{b}} while Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 using Macaulay2.^[19]^[20]^[21]

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, a primitive ideal of R is the annihilator of a (nonzero) simple R-module. The Jacobson radical Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J = \operatorname{Jac}(R)} of R is the intersection of all primitive ideals. Equivalently,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J = \bigcap_{\mathfrak{m} \text{ maximal ideals}} \mathfrak{m}.}

Indeed, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is a simple module and x is a nonzero element in M, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Rx = M} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R/\operatorname{Ann}(M) = R/\operatorname{Ann}(x) \simeq M} , meaning Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Ann}(M)} is a maximal ideal. Conversely, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{m}} is a maximal ideal, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{m}} is the annihilator of the simple R-module Template:Tmath. There is also another characterization (the proof is not hard):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 - yx} is a unit element if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Template:Tmath, then M does not admit a maximal submodule, since if there is a maximal submodule Template:Tmath, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J \cdot (M/L) = 0} and so Template:Tmath, a contradiction. Since a nonzero finitely generated module admits a maximal submodule, in particular, one has:

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle JM = M} and M is finitely generated, then Template:Tmath.

A maximal ideal is a prime ideal and so one has

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{nil}(R) = \bigcap_{\mathfrak{p} \text { prime ideals }} \mathfrak{p} \subset \operatorname{Jac}(R)}

where the intersection on the left is called the nilradical of R. As it turns out, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{nil}(R)} is also the set of nilpotent elements of R.

If R is an Artinian ring, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Jac}(R)} is nilpotent and Template:Tmath. (Proof: first note the DCC implies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J^n = J^{n+1}} for some n. If (DCC) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a} \supsetneq \operatorname{Ann}(J^n)} is an ideal properly minimal over the latter, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J \cdot (\mathfrak{a}/\operatorname{Ann}(J^n)) = 0} . That is, Template:Tmath, a contradiction.)

Extension and contraction of an ideal

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} be two commutative rings, and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: A\to B} be a ring homomorphism. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}} is an ideal in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(\mathfrak{a})} need not be an ideal in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} (e.g. take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} to be the inclusion of the ring of integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}} into the field of rationals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Q}} ). The extension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}^e} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} is defined to be the ideal in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} generated by Template:Tmath. Explicitly,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}^e = f(\mathfrak{a})B = \Big\{ \sum y_if(x_i) : x_i \in \mathfrak{a}, y_i \in B \Big\}.}

By abuse of notation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}B} is another common notation for this ideal extension.

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{b}} is an ideal of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(\mathfrak{b})} is always an ideal of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , called the contraction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{b}^c} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{b}} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} .

Assuming Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: A\to B} is a ring homomorphism, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}} is an ideal in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{b}} is an ideal in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} , then:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{b}} is prime in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{b}^c} is prime in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} ,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}^{ec} \supseteq \mathfrak{a},}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{b}^{ce} \subseteq \mathfrak{b}.}

It is false, in general, that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}} being prime (or maximal) in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} implies that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}^e} is prime (or maximal) in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} . Many classic examples of this stem from algebraic number theory. For example, consider the embedding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z} \to \mathbb{Z}\left\lbrack i \right\rbrack.} In Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = \mathbb{Z}\left\lbrack i \right\rbrack} , the element 2 factors as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 = (1 + i)(1 - i)} where (one can show) neither of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 + i, 1 - i} are units in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} . So Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2)^e} is not prime in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} (and therefore not maximal, as well). Indeed, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1 \pm i)^2 = \pm 2i} shows that Template:Tmath, Template:Tmath, and therefore Template:Tmath.

On the other hand, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is surjective and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a} \supseteq \ker f} then:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}^{ec}=\mathfrak{a} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{b}^{ce}=\mathfrak{b}, }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}} is a prime ideal in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftrightarrow} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}^e} is a prime ideal in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} ,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}} is a maximal ideal in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftrightarrow} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a}^e} is a maximal ideal in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} .

Remark: Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} be a field extension of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} , and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} be the rings of integers of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} , respectively. Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} is an integral extension of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , and we let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} be the inclusion map from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} . The behaviour of a prime ideal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{a} = \mathfrak{p}} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} under extension is one of the central problems of algebraic number theory.

The following is sometimes useful:^[22] a prime ideal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{p}} is a contraction of a prime ideal if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{p} = \mathfrak{p}^{ec}} . (Proof: Assuming the latter, note Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{p}^e B_{\mathfrak{p}} = B_{\mathfrak{p}} \Rightarrow \mathfrak{p}^e} intersects Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A - \mathfrak{p}} , a contradiction. Now, the prime ideals of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{\mathfrak{p}}} correspond to those in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} that are disjoint from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A - \mathfrak{p}} . Hence, there is a prime ideal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{q}} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} , disjoint from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A - \mathfrak{p}} , such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{q} B_{\mathfrak{p}}} is a maximal ideal containing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{p}^e B_{\mathfrak{p}}} . One then checks that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{q}} lies over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{p}} . The converse is obvious.)

Generalizations

Ideals can be generalized to any monoid object Template:Tmath, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is the object where the monoid structure has been forgotten. A left ideal of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a subobject Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} that "absorbs multiplication from the left by elements of Template:Tmath"; that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is a left ideal if it satisfies the following two conditions:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is a subobject of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R}
For every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \in (R,\otimes)} and every Template:Tmath, the product Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \otimes x} is in Template:Tmath.

A right ideal is defined with the condition "Template:Tmath" replaced by "'Template:Tmath". A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.

Notes

↑ Some authors call the zero and unit ideals of a ring R the trivial ideals of R.
↑ 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.

References

↑ ^1.0 ^1.1 John 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.
↑ ^9.0 ^9.1 ^9.2 ^9.3 Dummit & Foote (2004), p. 243.
↑ Lang (2005), Section III.2.
↑ Dummit & Foote (2004), p. 244.
↑ Because simple commutative rings are fields. See Lam (2001). [[[:Template:Google books]] A First Course in Noncommutative Rings] Check |url= value (help). 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 from the original on 2017-01-16. Retrieved 2017-01-14.
↑ "sums, products, and powers of ideals". www.math.uiuc.edu. Archived from the original on 2017-01-16. Retrieved 2017-01-14.
↑ "intersection of ideals". www.math.uiuc.edu. Archived from the original on 2017-01-16. Retrieved 2017-01-14.
↑ Atiyah & Macdonald (1969), Proposition 3.16.

Atiyah, Michael F.; Macdonald, Ian G. (1969). Introduction to Commutative Algebra. Perseus Books. ISBN 0-201-00361-9.
Dummit, David Steven; Foote, Richard Martin (2004). Abstract algebra (Third ed.). Hoboken, NJ: John Wiley & Sons, Inc. ISBN 9780471433347.
Eisenbud, David (1995), Commutative Algebra with a View toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-5350-1, ISBN 978-0-387-94268-1, MR 1322960
Lang, Serge (2005). Undergraduate Algebra (Third ed.). Springer-Verlag. ISBN 978-0-387-22025-3.
Hazewinkel, Michiel; Gubareni, Nadiya; Gubareni, Nadezhda Mikhaĭlovna; Kirichenko, Vladimir V. (2004). Algebras, rings and modules. 1. Springer. ISBN 1-4020-2690-0.
Milnor, John Willard (1971). Introduction to algebraic K-theory. Annals of Mathematics Studies. 72. Princeton, NJ: Princeton University Press. ISBN 9780691081014. MR 0349811. Zbl 0237.18005.

External links

Levinson, Jake (July 14, 2014). "The Geometric Interpretation for Extension of Ideals?". Stack Exchange.

[9] Some authors call the zero and unit ideals of a ring R the trivial ideals of R.

[13] 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.

[Stillwell-1] 1.0 ^1.1 John Stillwell (2010). Mathematics and its history. p. 439.

[flt-2] Harold M. Edwards (1977). Fermat's last theorem. A genetic introduction to algebraic number theory. p. 76.

[everest_ward-3] Everest G., Ward T. (2005). An introduction to number theory. p. 83.

[4] Dummit & Foote 2004, p. 242

[5] Dummit & Foote 2004, § 10.1., Examples (1).

[6] Dummit & Foote 2004, § 10.1., Proposition 3.

[7] Dummit & Foote 2004, Ch. 7, Proposition 6.

[8] Dummit & Foote 2004, Ch. 7, Theorem 7.

[FOOTNOTEDummitFoote2004243-10] 9.0 ^9.1 ^9.2 ^9.3 Dummit & Foote (2004), p. 243.

[FOOTNOTELang2005Section_III.2-11] Lang (2005), Section III.2.

[FOOTNOTEDummitFoote2004244-12] Dummit & Foote (2004), p. 244.

[14] Because simple commutative rings are fields. See Lam (2001). [[[:Template:Google books]] A First Course in Noncommutative Rings] Check |url= value (help). p. 39.

[15] "Zero ideal". Math World. 22 Aug 2024.

[FOOTNOTEDummitFoote2004255-16] Dummit & Foote (2004), p. 255.

[FOOTNOTEDummitFoote2004251-17] Dummit & Foote (2004), p. 251.

[Matsumura1-18] Matsumura, Hideyuki (1987). Commutative Ring Theory. Cambridge: Cambridge University Press. p. 132. ISBN 9781139171762.

[19] Eisenbud 1995, Exercise A 3.17

[FOOTNOTEMilnor19719-20] Milnor (1971), p. 9.

[21] "ideals". www.math.uiuc.edu. Archived from the original on 2017-01-16. Retrieved 2017-01-14.

[22] "sums, products, and powers of ideals". www.math.uiuc.edu. Archived from the original on 2017-01-16. Retrieved 2017-01-14.

[23] "intersection of ideals". www.math.uiuc.edu. Archived from the original on 2017-01-16. Retrieved 2017-01-14.

[FOOTNOTEAtiyahMacdonald1969Proposition_3.16-24] Atiyah & Macdonald (1969), Proposition 3.16.

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[19]

[20]

[21]

[22]

Ideal (ring theory)

Contents

History

Definitions

Notes on convention

Examples and properties

Types of ideals

Ideal operations

Examples of ideal operations

Radical of a ring

Extension and contraction of an ideal

Generalizations

See also

Notes

References

External links

Navigation menu

Ideal (ring theory)

History

Definitions

Notes on convention

Examples and properties

Types of ideals

Ideal operations

Examples of ideal operations

Radical of a ring

Extension and contraction of an ideal

Generalizations

See also

Notes

References

External links

Navigation menu

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