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{{Short description|Submodule of a mathematical ring}}
{{Ring theory sidebar|Basic}}
In [[mathematics]], and more specifically in [[ring theory]], an '''ideal''' of a [[ring (mathematics)|ring]] is a special [[subset]] of its elements. Ideals generalize certain subsets of the [[integer]]s, 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 (mathematics)|closure]] and [[Absorbing element|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 integer]]s: 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 ideal]]s of a ring are analogous to [[prime number]]s, and the [[Chinese remainder theorem]] can be generalized to ideals. There is a version of [[fundamental theorem of arithmetic|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 (order theory)|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 number]]s 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.<ref name="Stillwell">
{{cite book
| author = John Stillwell
| title = Mathematics and its history
| year = 2010
| page = 439
}}</ref>
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.<ref name="Stillwell" /><ref name="flt">
{{cite book
| author = Harold M. Edwards
| title = Fermat's last theorem. A genetic introduction to algebraic number theory
| year = 1977
| page = 76
}}</ref><ref name="everest_ward">
{{cite book
| author = Everest G., Ward T.
| title = An introduction to number theory
| year = 2005
| page = 83
}}</ref>
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 (mathematics)|ring]] {{mvar|R}}, a '''left ideal''' is a subset {{mvar|I}} of {{mvar|R}} that is a [[subgroup]] of the [[additive group]] of <math>R</math> that "absorbs multiplication from the left by elements of {{tmath|1= R }}"; that is, <math>I</math> is a left ideal if it satisfies the following two conditions:
# <math>(I,+)</math> is a [[subgroup]] of {{tmath|1= (R,+) }},
# For every <math>r \in R</math> and every {{tmath|1= x \in I }}, the product <math>r x</math> is in {{tmath|1= I }}.<ref>{{harvnb|Dummit|Foote|2004|p=242}}</ref>
In other words, a left ideal is a left [[submodule]] of {{mvar|R}}, considered as a [[left module]] over itself.<ref>{{harvnb|Dummit|Foote|2004|loc=§ 10.1., Examples (1).}}</ref>
A '''right ideal''' is defined similarly, with the condition <math>rx\in I</math> replaced by {{tmath|1= xr\in I }}. A '''two-sided ideal''' is a left ideal that is also a right ideal.
If the ring is [[commutative ring|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".
If {{mvar|I}} is a left, right or two-sided ideal, the relation
: <math>x \sim y \iff x-y\in I</math>
is an [[equivalence relation]] on {{mvar|R}}, and the set of [[equivalence class]]es forms a left, right or bi module denoted <math>R/I</math> and called the ''[[quotient module|quotient]]'' of {{mvar|R}} by {{mvar|I}}.<ref>{{harvnb|Dummit|Foote|2004|loc=§ 10.1., Proposition 3.}}</ref> (It is an instance of a [[congruence relation]] and is a generalization of [[modular arithmetic]].)
If the ideal {{mvar|I}} is two-sided, <math>R/I</math> is a ring,<ref>{{harvnb|Dummit|Foote|2004|loc=Ch. 7, Proposition 6.}}</ref> and the function
:<math>R\to R/I</math>
that associates to each element of {{mvar|R}} its equivalence class is a [[surjective]] [[ring homomorphism]] that has the ideal as its [[kernel (algebra)|kernel]].<ref>{{harvnb|Dummit|Foote|2004|loc=Ch. 7, Theorem 7.}}</ref> Conversely, the kernel of a ring homomorphism is a two-sided ideal. Therefore, ''the two-sided ideals are exactly the kernels of ring homomorphisms.''
===
By convention, a ring has the multiplicative identity. But some authors do not require a ring to have the multiplicative identity; i.e., for them, a ring is a [[rng (mathematics)|rng]]. For a rng {{mvar|R}}, a '''left ideal''' {{mvar|I}} is a {{not a typo|subrng}} with the additional property that <math>rx</math> is in {{mvar|I}} for every <math>r \in R</math> and every <math>x \in I</math>. (Right and two-sided ideals are defined similarly.) For a ring, an ideal {{mvar|I}} (say a left ideal) is rarely a subring; since a subring shares the same multiplicative identity with the ambient ring {{mvar|R}}, if {{mvar|I}} were a subring, for every <math>r \in R</math>, we have <math>r = r 1 \in I;</math> i.e., <math>I = R</math>.
The notion of an ideal does not involve associativity; thus, an ideal is also defined for [[non-associative ring]]s (often without the multiplicative identity) such as a [[Lie algebra]].
Traditionally, ideals are denoted using [[Fraktur]] lower-case letters, generally the first few letters (<math>\mathfrak{a},\mathfrak{b},\mathfrak{c}</math>, etc.) for generic ideals, <math>\mathfrak{m}</math> for maximal ideals, and <math>\mathfrak{p}</math> (and sometimes <math>\mathfrak{q}</math>) for prime ideals. In modern texts, capital letters, like <math>I</math> or <math>J</math> (or <math>M</math> and <math>P</math> 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 <math>(1)</math> since it is precisely the two-sided ideal generated (see below) by the unity {{tmath|1= 1_R }}. Also, the set <math>\{ 0_R \}</math> consisting of only the additive identity 0<sub>''R''</sub> forms a two-sided ideal called the '''zero ideal''' and is denoted by {{tmath|1= (0) }}.<ref group=note>Some authors call the zero and unit ideals of a ring ''R'' the '''trivial ideals''' of ''R''.</ref> Every (left, right or two-sided) ideal contains the zero ideal and is contained in the unit ideal.{{sfnp|Dummit|Foote|2004|p=243}}
* An (left, right or two-sided) ideal that is not the unit ideal is called a '''proper ideal''' (as it is a [[proper subset]]).{{sfnp|Lang|2005|loc=Section III.2}} Note: a left ideal <math>\mathfrak{a}</math> is proper if and only if it does not contain a unit element, since if <math>u \in \mathfrak{a}</math> is a unit element, then <math>r = (r u^{-1}) u \in \mathfrak{a}</math> for every {{tmath|1= r \in R }}. Typically there are plenty of proper ideals. In fact, if ''R'' is a [[skew-field]], then <math>(0), (1)</math> are its only ideals and conversely: that is, a nonzero ring ''R'' is a skew-field if <math>(0), (1)</math> are the only left (or right) ideals. (Proof: if <math>x</math> is a nonzero element, then the principal left ideal <math>Rx</math> (see below) is nonzero and thus <math>Rx = (1)</math>; i.e., <math>yx = 1</math> for some nonzero {{tmath|1= y }}. Likewise, <math>zy = 1</math> for some nonzero <math>z</math>. Then <math>z = z(yx) = (zy)x = x</math>.)
* The even [[integer]]s form an ideal in the ring <math>\mathbb{Z}</math> 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 {{tmath|1= 2\mathbb{Z} }}. More generally, the set of all integers divisible by a fixed integer <math>n</math> is an ideal denoted {{tmath|1= n\mathbb{Z} }}. In fact, every non-zero ideal of the ring <math>\mathbb{Z}</math> is generated by its smallest positive element, as a consequence of [[Euclidean division]], so <math>\mathbb{Z}</math> is a [[principal ideal ___domain]].{{sfnp|Dummit|Foote|2004|p=243}}
* The set of all [[polynomial]]s with real coefficients that are divisible by the polynomial <math>x^2+1</math> is an ideal in the ring of all real-coefficient polynomials {{tmath|1= \mathbb{R}[x] }}.
* Take a ring <math>R</math> and positive integer {{tmath|1= n }}. For each {{tmath|1= 1\leq i\leq n }}, the set of all <math>n\times n</math> [[matrix (mathematics)|matrices]] with entries in <math>R</math> whose <math>i</math>-th row is zero is a right ideal in the ring <math>M_n(R)</math> of all <math>n\times n</math> matrices with entries in {{tmath|1= R }}. It is not a left ideal. Similarly, for each {{tmath|1= 1\leq j\leq n }}, the set of all <math>n\times n</math> matrices whose <math>j</math>-th ''column'' is zero is a left ideal but not a right ideal.
* The ring <math>C(\mathbb{R})</math> of all [[continuous function]]s <math>f</math> from <math>\mathbb{R}</math> to <math>\mathbb{R}</math> under [[pointwise multiplication]] contains the ideal of all continuous functions <math>f</math> such that {{tmath|1= f(1)=0 }}.{{sfnp|Dummit|Foote|2004|p=244}} Another ideal in <math>C(\mathbb{R})</math> is given by those functions that vanish for large enough arguments, i.e. those continuous functions <math>f</math> for which there exists a number <math>L>0</math> such that <math>f(x)=0</math> whenever {{tmath|1= \vert x \vert >L }}.
* A ring is called a [[simple ring]] if it is nonzero and has no two-sided ideals other than {{tmath|1= (0), (1) }}. 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 <math>f: R \to S</math> is a [[ring homomorphism]], then the kernel <math>\ker(f) = f^{-1}(0_S)</math> is a two-sided ideal of {{tmath|1= R }}.{{sfnp|Dummit|Foote|2004|p=243}} By definition, {{tmath|1= f(1_R) = 1_S }}, and thus if <math>S</math> is not the [[zero ring]] (so {{tmath|1= 1_S\ne0_S }}), then <math>\ker(f)</math> is a proper ideal. More generally, for each left ideal ''I'' of ''S'', the pre-image <math>f^{-1}(I)</math> is a left ideal. If ''I'' is a left ideal of ''R'', then <math>f(I)</math> is a left ideal of the subring <math>f(R)</math> of ''S'': unless ''f'' is surjective, <math>f(I)</math> need not be an ideal of ''S''; see also {{slink|#Extension and contraction of an ideal}}.
* '''Ideal correspondence''': Given a surjective ring homomorphism {{tmath|1= f: R \to S }}, there is a bijective order-preserving correspondence between the left (resp. right, two-sided) ideals of <math>R</math> containing the kernel of <math>f</math> and the left (resp. right, two-sided) ideals of <math>S</math>: the correspondence is given by <math>I \mapsto f(I)</math> and the pre-image {{tmath|1= J \mapsto f^{-1}(J) }}. Moreover, for commutative rings, this bijective correspondence restricts to prime ideals, maximal ideals, and radical ideals (see the [[#Types_of_ideals|Types of ideals]] section for the definitions of these ideals).
* If ''M'' is a left ''R''-[[Module (mathematics)|module]] and <math>S \subset M</math> a subset, then the [[annihilator (ring theory)|annihilator]] <math>\operatorname{Ann}_R(S) = \{ r \in R \mid rs = 0, s \in S \}</math> of ''S'' is a left ideal. Given ideals <math>\mathfrak{a}, \mathfrak{b}</math> of a commutative ring ''R'', the ''R''-annihilator of <math>(\mathfrak{b} + \mathfrak{a})/\mathfrak{a}</math> is an ideal of ''R'' called the [[ideal quotient]] of <math>\mathfrak{a}</math> by <math>\mathfrak{b}</math> and is denoted by {{tmath|1= (\mathfrak{a} : \mathfrak{b}) }}; it is an instance of [[idealizer]] in commutative algebra.
* Let <math>\mathfrak{a}_i, i \in S</math> be an '''[[ascending chain]] of left ideals''' in a ring ''R''; i.e., <math>S</math> is a totally ordered set and <math>\mathfrak{a}_i \subset \mathfrak{a}_j</math> for each {{tmath|1= i < j }}. Then the union <math>\textstyle \bigcup_{i \in S} \mathfrak{a}_i</math> 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 <math>E \subset R</math> is a possibly empty subset and <math>\mathfrak{a}_0 \subset R</math> is a left ideal that is disjoint from ''E'', then there is an ideal that is maximal among the ideals containing <math>\mathfrak{a}_0</math> and disjoint from ''E''. (Again this is still valid if the ring ''R'' lacks the unity 1.) When <math>R \ne 0</math>, taking <math>\mathfrak{a}_0 = (0)</math> and {{tmath|1= E = \{ 1 \} }}, in particular, there exists a left ideal that is maximal among proper left ideals (often simply called a maximal left ideal); see [[Krull's theorem]] for more.
*An arbitrary union of ideals need not be an ideal, but the following is still true: given a possibly empty subset ''X'' of ''R'', there is the smallest left ideal containing ''X'', called the left ideal generated by ''X'' and is denoted by {{tmath|1= RX }}. Such an ideal exists since it is the intersection of all left ideals containing ''X''. Equivalently, <math>RX</math> is the set of all the [[linear combinations|(finite) left ''R''-linear combinations]] of elements of ''X'' over ''R'': {{br}}<math display="block">RX = \{r_1x_1+\dots+r_nx_n \mid n\in\mathbb{N}, r_i\in R, x_i\in X\}</math>(since such a span is the smallest left ideal containing ''X''.)<ref group = note>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 {{nowrap|''x'' + ''x'' + ... + ''x''}}, and ''n''-fold sums of the form {{nowrap|(−''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.</ref> 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., {{br}}<math display="block">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\} .</math>
* 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 <math>Rx</math> (resp. {{tmath|1= xR, RxR }}). The principal two-sided ideal <math>RxR</math> is often also denoted by {{tmath|1= (x) }}. If <math>X = \{ x_1, \dots, x_n \}</math> is a [[finite set]], then <math>RXR</math> is also written as {{tmath|1= (x_1, \dots, x_n) }}.
* There is a bijective correspondence between ideals and [[congruence relation]]s (equivalence relations that respect the ring structure) on the ring: Given an ideal <math>I</math> of a ring {{tmath|1= R }}, let <math>x\sim y</math> if {{tmath|1= x-y\in I }}. Then <math>\sim</math> is a congruence relation on {{tmath|1= R }}. Conversely, given a congruence relation <math>\sim</math> on {{tmath|1= R }}, let {{tmath|1= I=\{x\in R:x\sim 0\} }}. Then <math>I</math> is an ideal of {{tmath|1= R }}.
== Types of ideals ==
''To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles.''
Ideals are important because they appear as kernels of ring homomorphisms and allow one to define [[factor ring]]s. Different types of ideals are studied because they can be used to construct different types of factor rings.
* '''[[Maximal ideal]]''': A proper ideal {{mvar|I}} is called a '''maximal ideal''' if there exists no other proper ideal {{math|''J''}} with {{mvar|I}} a proper subset of {{math|''J''}}. The factor ring of a maximal ideal is a [[simple ring]] in general and is a [[field (mathematics)|field]] for commutative rings.<ref>Because simple commutative rings are fields. See {{cite book|author=Lam|year=2001|title=A First Course in Noncommutative Rings|url={{Google books|plainurl=y|id=f15FyZuZ3-4C|page=39|text=simple commutative rings}}|page=39}}</ref>
* '''[[Minimal ideal]]''': A nonzero ideal is called minimal if it contains no other nonzero ideal.
* '''Zero ideal''': the ideal <math>\{0\}</math>.<ref>{{cite web|url=https://mathworld.wolfram.com/ZeroIdeal.html|title=Zero ideal|website=Math World|date=22 Aug 2024}}</ref>
* '''Unit ideal''': the whole ring (being the ideal generated by <math>1</math>).{{sfnp|Dummit|Foote|2004|p=243}}
* '''[[Prime ideal]]''': A proper ideal <math>I</math> is called a '''prime ideal''' if for any <math>a</math> and <math>b</math> in {{tmath|1= R }}, if <math>ab</math> is in {{tmath|1= I }}, then at least one of <math>a</math> and <math>b</math> is in {{tmath|1= I }}. The factor ring of a prime ideal is a [[prime ring]] in general and is an [[integral ___domain]] for commutative rings.{{sfnp|Dummit|Foote|2004|p=255}}
* '''[[Radical of an ideal|Radical ideal]]''' or [[semiprime ideal]]: A proper ideal {{mvar|I}} is called '''radical''' or '''semiprime''' if for any {{math|''a''}} in <math>R</math>, if {{math|''a''<sup>''n''</sup>}} is in {{mvar|I}} for some {{math|''n''}}, then {{math|''a''}} is in {{mvar|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 {{mvar|I}} is called a '''primary ideal''' if for all {{math|''a''}} and {{math|''b''}} in {{math|''R''}}, if {{math|''ab''}} is in {{mvar|I}}, then at least one of {{math|''a''}} and {{math|''b''<sup>''n''</sup>}} is in {{mvar|I}} for some [[natural number]] {{math|''n''}}. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime.
* '''[[Principal ideal]]''': An ideal generated by ''one'' element.{{sfnp|Dummit|Foote|2004|p=251}}
* {{anchor|Finitely generated ideal}}'''Finitely generated ideal''': This type of ideal is [[finitely generated module|finitely generated]] as a module.
* '''[[Primitive ideal]]''': A left primitive ideal is the [[Annihilator (ring theory)|annihilator]] of a [[simple module|simple]] left [[module (mathematics)|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 {{mvar|I}}, {{mvar|J}} are said to be '''comaximal''' if <math>x + y = 1</math> for some <math>x \in I</math> and {{tmath|1= 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 a [[system of parameters]].
* '''[[Perfect ideal]]''': A proper ideal {{mvar|I}} in a Noetherian ring <math>R</math> is called a '''perfect ideal''' if its [[Grade (ring theory)|grade]] equals the [[projective dimension]] of the associated quotient ring,<ref name=Matsumura1>{{cite book |last=Matsumura |first=Hideyuki |author-link=Hideyuki Matsumura |date=1987 |title=Commutative Ring Theory |url=https://www.cambridge.org/core/books/commutative-ring-theory/02819830750568B06C16E6199F3562C1 |___location=Cambridge |publisher=Cambridge University Press | page=132 |isbn=9781139171762}}</ref> {{tmath|1= \textrm{grade}(I)=\textrm{proj}\dim(R/I) }}. A perfect ideal is [[Unmixed ideal|unmixed]].
* '''[[Cohen–Macaulay_ring#unmixed|Unmixed ideal]]''': A proper ideal {{mvar|I}} in a Noetherian ring <math>R</math> is called an '''unmixed ideal''' (in height) if the height of {{mvar|I}} is equal to the height of every [[associated prime]] {{math|''P''}} of <math>R/I</math>. (This is stronger than saying that <math>R/I</math> is [[equidimensionality|equidimensional]]. See also [[Minimal_prime_ideal#Equidimensional_ring|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 <math>R</math> is a commutative ___domain with [[quotient field]] <math>K</math>. Despite their names, fractional ideals are not necessarily ideals. A fractional ideal of <math>R</math> is an <math>R</math>-submodule {{tmath|1= I }} of <math>K</math> for which there exists a non-zero <math>r \in R</math> such that <math>rI \subseteq R</math>. If the fractional ideal is contained entirely in <math>R</math>, then it is truly an ideal of <math>R</math>.
* '''[[Invertible ideal]]''': Usually an invertible ideal {{math|''A''}} is defined as a fractional ideal for which there is another fractional ideal {{math|''B''}} such that {{math|1=''AB'' = ''BA'' = ''R''}}. Some authors may also apply "invertible ideal" to ordinary ring ideals {{math|''A''}} and {{math|''B''}} with {{math|1=''AB'' = ''BA'' = ''R''}} in rings other than domains.
== Ideal operations ==
The sum and product of ideals are defined as follows. For <math>\mathfrak{a}</math> and {{tmath|1= \mathfrak{b} }}, left (resp. right) ideals of a ring ''R'', their sum is
: <math>\mathfrak{a}+\mathfrak{b}:=\{a+b \mid a \in \mathfrak{a} \mbox{ and } b \in \mathfrak{b}\}</math>,
which is a left (resp. right) ideal,
and, if <math>\mathfrak{a}, \mathfrak{b}</math> are two-sided,
: <math>\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\},</math>
i.e. the product is the ideal generated by all products of the form ''ab'' with ''a'' in <math>\mathfrak{a}</math> and ''b'' in {{tmath|1= \mathfrak{b} }}.
Note <math>\mathfrak{a} + \mathfrak{b}</math> is the smallest left (resp. right) ideal containing both <math>\mathfrak{a}</math> and <math>\mathfrak{b}</math> (or the union {{tmath|1= \mathfrak{a} \cup \mathfrak{b} }}), while the product <math>\mathfrak{a}\mathfrak{b}</math> is contained in the intersection of <math>\mathfrak{a}</math> and {{tmath|1= \mathfrak{b} }}.
The distributive law holds for two-sided ideals {{tmath|1= \mathfrak{a}, \mathfrak{b}, \mathfrak{c} }},
* {{tmath|1= \mathfrak{a}(\mathfrak{b} + \mathfrak{c}) = \mathfrak{a} \mathfrak{b} + \mathfrak{a} \mathfrak{c} }},
* {{tmath|1= (\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:
: <math>\mathfrak{a} \cap (\mathfrak{b} + \mathfrak{c}) \supset \mathfrak{a} \cap \mathfrak{b} + \mathfrak{a} \cap \mathfrak{c}</math>
where the equality holds if <math>\mathfrak{a}</math> contains <math>\mathfrak{b}</math> or <math>\mathfrak{c}</math>.
'''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 lattice|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 <math>\mathfrak{a}, \mathfrak{b}</math> are ideals of a commutative ring ''R'', then <math>\mathfrak{a} \cap \mathfrak{b} = \mathfrak{a} \mathfrak{b}</math> in the following two cases (at least)
* <math>\mathfrak{a} + \mathfrak{b} = (1)</math>
* <math>\mathfrak{a}</math> is generated by elements that form a [[regular sequence]] modulo {{tmath|1= \mathfrak{b} }}.
(More generally, the difference between a product and an intersection of ideals is measured by the [[Tor functor]]: {{tmath|1= \operatorname{Tor}^R_1(R/\mathfrak{a}, R/\mathfrak{b}) = (\mathfrak{a} \cap \mathfrak{b})/ \mathfrak{a} \mathfrak{b} }}.<ref>{{harvnb|Eisenbud|1995|loc=Exercise A 3.17}}</ref>)
An integral ___domain is called a [[Dedekind ___domain]] if for each pair of ideals <math>\mathfrak{a} \subset \mathfrak{b}</math>, there is an ideal <math>\mathfrak{c}</math> such that {{tmath|1= \mathfrak \mathfrak{a} = \mathfrak{b} \mathfrak{c} }}.{{sfnp|Milnor|1971|p=9}} 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 <math>\mathbb{Z}</math> we have
: <math>(n)\cap(m) = \operatorname{lcm}(n,m)\mathbb{Z}</math>
since <math>(n)\cap(m)</math> is the set of integers that are divisible by both <math>n</math> and {{tmath|1= m }}.
Let <math>R = \mathbb{C}[x,y,z,w]</math> and let {{tmath|1= \mathfrak{a} = (z, w), \mathfrak{b} = (x+z,y+w),\mathfrak{c} = (x+z, w) }}. Then,
* <math> \mathfrak{a} + \mathfrak{b} = (z,w, x+z, y+w) = (x, y, z, w)</math> and <math>\mathfrak{a} + \mathfrak{c} = (z, w, x)</math>
* <math>\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)</math>
* <math>\mathfrak{a}\mathfrak{c} = (xz + z^2, zw, xw + zw, w^2)</math>
* <math>\mathfrak{a} \cap \mathfrak{b} = \mathfrak{a}\mathfrak{b}</math> while <math>\mathfrak{a} \cap \mathfrak{c} = (w, xz + z^2) \neq \mathfrak{a}\mathfrak{c}</math>
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]].<ref>{{Cite web|url=http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_ideals.html|title=ideals|website=www.math.uiuc.edu|access-date=2017-01-14|archive-url=https://web.archive.org/web/20170116190119/http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_ideals.html|archive-date=2017-01-16|url-status=dead}}</ref><ref>{{Cite web|url=http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_sums_cm_spproducts_cm_spand_sppowers_spof_spideals.html|title=sums, products, and powers of ideals|website=www.math.uiuc.edu|access-date=2017-01-14|archive-url=https://web.archive.org/web/20170116185903/http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_sums_cm_spproducts_cm_spand_sppowers_spof_spideals.html|archive-date=2017-01-16|url-status=dead}}</ref><ref>{{Cite web|url=http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_intersection_spof_spideals.html|title=intersection of ideals|website=www.math.uiuc.edu|access-date=2017-01-14|archive-url=https://web.archive.org/web/20170116185829/http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_intersection_spof_spideals.html|archive-date=2017-01-16|url-status=dead}}</ref>
== Radical of a ring ==
{{main|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 module|simple ''R''-module]]. The [[Jacobson radical]] <math>J = \operatorname{Jac}(R)</math> of ''R'' is the intersection of all primitive ideals. Equivalently,
: <math>J = \bigcap_{\mathfrak{m} \text{ maximal ideals}} \mathfrak{m}.</math>
Indeed, if <math>M</math> is a simple module and ''x'' is a nonzero element in ''M'', then <math>Rx = M</math> and <math>R/\operatorname{Ann}(M) = R/\operatorname{Ann}(x) \simeq M</math>, meaning <math>\operatorname{Ann}(M)</math> is a maximal ideal. Conversely, if <math>\mathfrak{m}</math> is a maximal ideal, then <math>\mathfrak{m}</math> is the annihilator of the simple ''R''-module {{tmath|1= R/\mathfrak{m} }}. There is also another characterization (the proof is not hard):
: <math>J = \{ x \in R \mid 1 - yx \, \text{ is a unit element for every } y \in R\}.</math>
For a not-necessarily-commutative ring, it is a general fact that <math>1 - yx</math> is a [[unit element]] if and only if <math>1 - xy</math> 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 {{tmath|1= JM = M }}, then ''M'' does not admit a [[maximal submodule]], since if there is a maximal submodule {{tmath|1= L \subsetneq M }}, <math>J \cdot (M/L) = 0</math> and so {{tmath|1= M = JM \subset L \subsetneq M }}, a contradiction. Since a nonzero [[finitely generated module]] admits a maximal submodule, in particular, one has:
: If <math>JM = M</math> and ''M'' is finitely generated, then {{tmath|1= M = 0 }}.
A maximal ideal is a prime ideal and so one has
: <math>\operatorname{nil}(R) =
\bigcap_{\mathfrak{p} \text { prime ideals }} \mathfrak{p} \subset \operatorname{Jac}(R)</math>
where the intersection on the left is called the [[Nilradical of a ring|nilradical]] of ''R''. As it turns out, <math>\operatorname{nil}(R)</math> is also the set of [[nilpotent element]]s of ''R''.
If ''R'' is an [[Artinian ring]], then <math>\operatorname{Jac}(R)</math> is nilpotent and {{tmath|1= \operatorname{nil}(R) = \operatorname{Jac}(R) }}. (Proof: first note the DCC implies <math>J^n = J^{n+1}</math> for some ''n''. If (DCC) <math>\mathfrak{a} \supsetneq \operatorname{Ann}(J^n)</math> is an ideal properly minimal over the latter, then <math>J \cdot (\mathfrak{a}/\operatorname{Ann}(J^n)) = 0</math>. That is, {{tmath|1= J^n \mathfrak{a} = J^{n+1} \mathfrak{a} = 0 }}, a contradiction.)
== Extension and contraction of an ideal ==
Let ''A'' and ''B'' be two [[commutative ring]]s, and let ''f'' : ''A'' → ''B'' be a [[ring homomorphism]]. If <math>\mathfrak{a}</math> is an ideal in ''A'', then <math>f(\mathfrak{a})</math> need not be an ideal in ''B'' (e.g. take ''f'' to be the [[inclusion map|inclusion]] of the ring of integers '''Z''' into the field of rationals '''Q'''). The '''extension''' <math>\mathfrak{a}^e</math> of <math>\mathfrak{a}</math> in ''B'' is defined to be the ideal in ''B'' generated by {{tmath|1= f(\mathfrak{a}) }}. Explicitly,
: <math>\mathfrak{a}^e = \Big\{ \sum y_if(x_i) : x_i \in \mathfrak{a}, y_i \in B \Big\}</math>
If <math>\mathfrak{b}</math> is an ideal of ''B'', then <math>f^{-1}(\mathfrak{b})</math> is always an ideal of ''A'', called the '''contraction''' <math>\mathfrak{b}^c</math> of <math>\mathfrak{b}</math> to ''A''.
Assuming ''f'' : ''A'' → ''B'' is a ring homomorphism, <math>\mathfrak{a}</math> is an ideal in ''A'', <math>\mathfrak{b}</math> is an ideal in ''B'', then:
* <math>\mathfrak{b}</math> is prime in ''B'' <math>\Rightarrow</math> <math>\mathfrak{b}^c</math> is prime in ''A''.
* <math>\mathfrak{a}^{ec} \supseteq \mathfrak{a}</math>
* <math>\mathfrak{b}^{ce} \subseteq \mathfrak{b}</math>
It is false, in general, that <math>\mathfrak{a}</math> being prime (or maximal) in ''A'' implies that <math>\mathfrak{a}^e</math> is prime (or maximal) in ''B''. Many classic examples of this stem from algebraic number theory. For example, [[embedding]] {{tmath|1= \mathbb{Z} \to \mathbb{Z}\left\lbrack i \right\rbrack }}. In <math>B = \mathbb{Z}\left\lbrack i \right\rbrack</math>, the element 2 factors as <math>2 = (1 + i)(1 - i)</math> where (one can show) neither of <math>1 + i, 1 - i</math> are units in ''B''. So <math>(2)^e</math> is not prime in ''B'' (and therefore not maximal, as well). Indeed, <math>(1 \pm i)^2 = \pm 2i</math> shows that {{tmath|1= (1 + i) = ((1 - i) - (1 - i)^2) }}, {{tmath|1= (1 - i) = ((1 + i) - (1 + i)^2) }}, and therefore {{tmath|1= (2)^e = (1 + i)^2 }}.
On the other hand, if ''f'' is [[surjective_function|surjective]] and [[kernel_(algebra)|<math> \mathfrak{a} \supseteq \ker f</math>]] then:
* <math>\mathfrak{a}^{ec}=\mathfrak{a} </math> and {{tmath|1= \mathfrak{b}^{ce}=\mathfrak{b} }}.
* <math>\mathfrak{a}</math> is a [[prime ideal]] in ''A'' <math>\Leftrightarrow</math> <math>\mathfrak{a}^e</math> is a prime ideal in ''B''.
* <math>\mathfrak{a}</math> is a [[maximal ideal]] in ''A'' <math>\Leftrightarrow</math> <math>\mathfrak{a}^e</math> is a maximal ideal in ''B''.
'''Remark''': Let ''K'' be a [[field extension]] of ''L'', and let ''B'' and ''A'' be the [[ring of integers|rings of integers]] of ''K'' and ''L'', respectively. Then ''B'' is an [[integral extension]] of ''A'', and we let ''f'' be the [[inclusion map]] from ''A'' to ''B''. The behaviour of a [[prime ideal]] <math>\mathfrak{a} = \mathfrak{p}</math> of ''A'' under extension is one of the central problems of [[algebraic number theory]].
The following is sometimes useful:{{sfnp|Atiyah|Macdonald|1969|loc=Proposition 3.16}} a prime ideal <math>\mathfrak{p}</math> is a contraction of a prime ideal if and only if {{tmath|1= \mathfrak{p} = \mathfrak{p}^{ec} }}. (Proof: Assuming the latter, note <math>\mathfrak{p}^e B_{\mathfrak{p}} = B_{\mathfrak{p}} \Rightarrow \mathfrak{p}^e</math> intersects {{tmath|1= A - \mathfrak{p} }}, a contradiction. Now, the prime ideals of <math>B_{\mathfrak{p}}</math> correspond to those in ''B'' that are disjoint from {{tmath|1= A - \mathfrak{p} }}. Hence, there is a prime ideal <math>\mathfrak{q}</math> of ''B'', disjoint from {{tmath|1= A - \mathfrak{p} }}, such that <math>\mathfrak{q} B_{\mathfrak{p}}</math> is a maximal ideal containing {{tmath|1= \mathfrak{p}^e B_{\mathfrak{p} } }}. One then checks that <math>\mathfrak{q}</math> lies over {{tmath|1= \mathfrak{p} }}. The converse is obvious.)
== Generalizations ==
Ideals can be generalized to any [[monoid object]] {{tmath|1= (R,\otimes) }}, where <math>R</math> is the object where the [[monoid]] structure has been [[Forgetful functor|forgotten]]. A '''left ideal''' of <math>R</math> is a [[subobject]] <math>I</math> that "absorbs multiplication from the left by elements of {{tmath|1= R }}"; that is, <math>I</math> is a '''left ideal''' if it satisfies the following two conditions:
# <math>I</math> is a [[subobject]] of <math>R</math>
# For every <math>r \in (R,\otimes)</math> and every {{tmath|1= x \in (I, \otimes) }}, the product <math>r \otimes x</math> is in {{tmath|1= (I, \otimes) }}.
A '''right ideal''' is defined with the condition "{{tmath|1= r \otimes x \in (I, \otimes) }}" replaced by "'{{tmath|1= x \otimes r \in (I, \otimes) }}". A '''two-sided ideal''' is a left ideal that is also a right ideal, and is sometimes simply called an ideal. When <math>R</math> is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the term '''ideal''' is used alone.<!-- This section is about generalization so it’s weird to have general facts on usual ideals.
An ideal can also be thought of as a specific type of [[Module_(mathematics)|{{math|''R''}}-module]]. If we consider <math>R</math> as a left <math>R</math>-module (by left multiplication), then a left ideal <math>I</math> is really just a left [[Module_(mathematics)#Submodules_and_homomorphisms|sub-module]] of {{tmath|1= R }}. In other words, <math>I</math> is a left (right) ideal of <math>R</math> if and only if it is a left (right) <math>R</math>-module that is a subset of {{tmath|1= R }}. <math>I</math> is a two-sided ideal if it is a sub-<math>R</math>-bimodule of {{tmath|1= R }}.
Example: If we let {{tmath|1= R=\Z }}, an ideal of <math>\mathbb{Z}</math> is a subset of {{tmath|1= \Z }} that is an abelian group under addition, i.e. it has the form {{tmath|1= m\Z }} for some {{tmath|1= m\in\Z }}. So these give all the ideals of {{tmath|1= \Z }}.-->
== See also ==
* [[Modular arithmetic]]
* [[Noether isomorphism theorem]]
* [[Boolean prime ideal theorem]]
* [[Ideal theory]]
* [[Ideal (order theory)]]
* [[Ideal norm]]
* [[Splitting of prime ideals in Galois extensions]]
* [[Ideal sheaf]]
== Notes ==
{{reflist|group=note}}
== References ==
{{reflist}}
{{refbegin}}
* {{cite book |last1=Atiyah |first1=Michael F. |authorlink1=Michael Atiyah |last2=Macdonald |first2=Ian G. |authorlink2=Ian G. Macdonald |title=[[Introduction to Commutative Algebra]] |publisher=Perseus Books |year=1969 |isbn=0-201-00361-9}}
* {{cite book |last1=Dummit |first1=David Steven |last2=Foote |first2=Richard Martin |title=Abstract algebra |date=2004 |publisher=John Wiley & Sons, Inc. |___location=Hoboken, NJ |isbn=9780471433347 |edition=Third}}
* {{citation |last1=Eisenbud |first1=David |author1-link=David Eisenbud |title=Commutative Algebra with a View toward Algebraic Geometry |publisher=[[Springer-Verlag]] |___location=Berlin, New York |series=[[Graduate Texts in Mathematics]] |isbn=978-0-387-94268-1 |mr=1322960 |year=1995 |volume=150 |doi=10.1007/978-1-4612-5350-1}}
* {{cite book |last=Lang |first=Serge | author-link=Serge Lang | title=Undergraduate Algebra | edition=Third | publisher=[[Springer-Verlag]] | isbn=978-0-387-22025-3 | year=2005 }}
* {{cite book |first1=Michiel |last1=Hazewinkel |authorlink1=Michiel Hazewinkel |first2=Nadiya |last2=Gubareni |first3=Nadezhda Mikhaĭlovna |last3=Gubareni |first4=Vladimir V. |last4=Kirichenko |title=Algebras, rings and modules |volume=1 |publisher=Springer |year=2004 |isbn=1-4020-2690-0}}
* {{cite book |last1=Milnor |first1=John Willard |author1-link=John Milnor |title=Introduction to algebraic K-theory |series=Annals of Mathematics Studies |volume=72 |publisher=[[Princeton University Press]] |___location=Princeton, NJ |year=1971 |isbn=9780691081014 |mr=0349811 |zbl=0237.18005}}
{{refend}}
== External links ==
* {{cite web |first=Jake |last=Levinson |title=The Geometric Interpretation for Extension of Ideals? |date=July 14, 2014 |url=https://math.stackexchange.com/q/866598 |work=[[Stack Exchange]] }}
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[[Category:Ideals (ring theory)| ]]
[[Category:Algebraic structures]]
[[Category:Commutative algebra]]
[[Category:Algebraic number theory]]
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