Ideal (ring theory)

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In ring theory, a branch of abstract algebra, 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 other integer results in another even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group.

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Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals. There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory).

The related, but distinct, concept of an ideal in order theory is derived from the notion of ideal in ring theory. A fractional ideal is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity.

History

Ernst Kummer invented the concept of ideal numbers to serve as the "missing" factors in number rings in which unique factorization fails; here the word "ideal" is in the sense of existing in imagination only, in analogy with "ideal" objects in geometry such as points at infinity. [1] In 1876, Richard Dedekind replaced Kummer's undefined concept by concrete sets of numbers, sets that he called ideals, in the third edition of Dirichlet's book Vorlesungen über Zahlentheorie , to which Dedekind had added many supplements. [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 and motivation

For an arbitrary ring ${\displaystyle (R,+,\cdot )}$, let ${\displaystyle (R,+)}$ be its additive group. A subset ${\displaystyle I}$ is called a left ideal of ${\displaystyle R}$ if it is an additive subgroup of ${\displaystyle R}$ that "absorbs multiplication from the left by elements of ${\displaystyle R}$"; that is, ${\displaystyle I}$ is a left ideal if it satisfies the following two conditions:

1. ${\displaystyle (I,+)}$ is a subgroup of ${\displaystyle (R,+),}$
2. For every ${\displaystyle r\in R}$ and every ${\displaystyle x\in I}$, the product ${\displaystyle rx}$ is in ${\displaystyle I}$.

A right ideal is defined with the condition "r xI" replaced by "x rI". A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. In the language of modules, the definitions mean that a left (resp. right, two-sided) ideal of R is precisely a left (resp. right, bi-) R-submodule of R when R is viewed as an R-module. When R is a commutative ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.

To understand the concept of an ideal, consider how ideals arise in the construction of rings of "elements modulo". For concreteness, let us look at the ring ℤn of integers modulo a given integer n ∈ ℤ (note that ℤ is a commutative ring). The key observation here is that we obtain ℤn by taking the integer line ℤ and wrapping it around itself so that various integers get identified. In doing so, we must satisfy two requirements: 1) n must be identified with 0 since n is congruent to 0 modulo n, and 2) the resulting structure must again be a ring. The second requirement forces us to make additional identifications (i.e., it determines the precise way in which we must wrap ℤ around itself). The notion of an ideal arises when we ask the question:

What is the exact set of integers that we are forced to identify with 0?

The answer is, unsurprisingly, the set nℤ = { nm | m ∈ ℤ } of all integers congruent to 0 modulo n. That is, we must wrap ℤ around itself infinitely many times so that the integers ..., n ⋅ (−2), n ⋅ (−1), n ⋅ (+1), n ⋅ (+2), ... will all align with 0. If we look at what properties this set must satisfy in order to ensure that ℤn is a ring, then we arrive at the definition of an ideal. Indeed, one can directly verify that nℤ is an ideal of ℤ.

Remark. Identifications with elements other than 0 also need to be made. For example, the elements in 1 + n must be identified with 1, the elements in 2 + n must be identified with 2, and so on. Those, however, are uniquely determined by nℤ since ℤ is an additive group.

We can make a similar construction in any commutative ring R: start with an arbitrary xR, and then identify with 0 all elements of the ideal xR = { x r : rR }. It turns out that the ideal xR is the smallest ideal that contains x, called the ideal generated by x. More generally, we can start with an arbitrary subset SR, and then identify with 0 all the elements in the ideal generated by S: the smallest ideal (S) such that S ⊆ (S). The ring that we obtain after the identification depends only on the ideal (S) and not on the set S that we started with. That is, if (S) = (T), then the resulting rings will be the same.

Therefore, an ideal I of a commutative ring R captures canonically the information needed to obtain the ring of elements of R modulo a given subset SR. The elements of I, by definition, are those that are congruent to zero, that is, identified with zero in the resulting ring. The resulting ring is called the quotient of R by I and is denoted R/I. Intuitively, the definition of an ideal postulates two natural conditions necessary for I to contain all elements designated as "zeros" by R/I:

1. I is an additive subgroup of R: the zero 0 of R is a "zero" 0 ∈ I, and if x1I and x2I are "zeros", then x1x2I is a "zero" too.
2. Any rR multiplied by a "zero" xI is a "zero" rxI.

It turns out that the above conditions are also sufficient for I to contain all the necessary "zeros": no other elements have to be designated as "zero" in order to form R/I. (In fact, no other elements should be designated as "zero" if we want to make the fewest identifications.)

Remark. The above construction still works using two-sided ideals even if R is not necessarily commutative.

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 ${\displaystyle (1)}$ since it is precisely the two-sided ideal generated (see below) by the unity ${\displaystyle 1_{R}}$. Also, the set ${\displaystyle \{0_{R}\}}$ consisting of only the additive identity 0R forms a two-sided ideal called the zero ideal and is denoted by ${\displaystyle (0)}$. [note 1] Every (left, right or two-sided) ideal contains the zero ideal and is contained in the unit ideal.
• An (left, right or two-sided) ideal that is not the unit ideal is called a proper ideal (as it is a proper subset). [4] Note: a left ideal ${\displaystyle {\mathfrak {a}}}$ is proper if and only if it does not contain a unit element, since if ${\displaystyle u\in {\mathfrak {a}}}$ is a unit element, then ${\displaystyle r=(ru^{-1})u\in {\mathfrak {a}}}$ for every ${\displaystyle r\in R}$. Typically there are plenty of proper ideals. In fact, if R is a skew-field, then ${\displaystyle (0),(1)}$ are its only ideals and conversely: that is, a nonzero ring R is a skew-field if ${\displaystyle (0),(1)}$ are the only left (or right) ideals. (Proof: if ${\displaystyle x}$ is a nonzero element, then the principal left ideal ${\displaystyle Rx}$ (see below) is nonzero and thus ${\displaystyle Rx=(1)}$; i.e., ${\displaystyle yx=1}$ for some nonzero ${\displaystyle y}$. Likewise, ${\displaystyle zy=1}$ for some nonzero ${\displaystyle z}$. Then ${\displaystyle z=z(yx)=(zy)x=x}$.)
• The even integers form an ideal in the ring ${\displaystyle \mathbb {Z} }$ of all integers; it is usually denoted by ${\displaystyle 2\mathbb {Z} }$. This is because the sum of any even integers is even, and the product of any integer with an even integer is also even. Similarly, the set of all integers divisible by a fixed integer n is an ideal denoted ${\displaystyle n\mathbb {Z} }$.
• The set of all polynomials with real coefficients which are divisible by the polynomial x2 + 1 is an ideal in the ring of all polynomials.
• The set of all n-by-n matrices whose last row is zero forms a right ideal in the ring of all n-by-n matrices. It is not a left ideal. The set of all n-by-n matrices whose last column is zero forms a left ideal but not a right ideal.
• The ring ${\displaystyle C(\mathbb {R} )}$ of all continuous functions f from ${\displaystyle \mathbb {R} }$ to ${\displaystyle \mathbb {R} }$ under pointwise multiplication contains the ideal of all continuous functions f such that f(1) = 0. Another ideal in ${\displaystyle C(\mathbb {R} )}$ is given by those functions which vanish for large enough arguments, i.e. those continuous functions f for which there exists a number L > 0 such that f(x) = 0 whenever |x| > L.
• A ring is called a simple ring if it is nonzero and has no two-sided ideals other than ${\displaystyle (0),(1)}$. 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 ${\displaystyle f:R\to S}$ is a ring homomorphism, then the kernel ${\displaystyle \ker(f)=f^{-1}(0_{S})}$ is a two-sided ideal of ${\displaystyle R}$. By definition, ${\displaystyle f(1_{R})=1_{S}}$, and thus if ${\displaystyle S}$ is not the zero ring (so ${\displaystyle 1_{S}\neq 0_{S}}$), then ${\displaystyle \ker(f)}$ is a proper ideal. More generally, for each left ideal I of S, the pre-image ${\displaystyle f^{-1}(I)}$ is a left ideal. If I is a left ideal of R, then ${\displaystyle f(I)}$ is a left ideal of the subring ${\displaystyle f(R)}$ of S: unless f is surjective, ${\displaystyle f(I)}$ need not be an ideal of S; see also #Extension and contraction of an ideal below.
• Ideal correspondence: Given a surjective ring homomorphism ${\displaystyle f:R\to S}$, there is a bijective order-preserving correspondence between the left (resp. right, two-sided) ideals of ${\displaystyle R}$ containing the kernel of ${\displaystyle f}$ and the left (resp. right, two-sided) ideals of ${\displaystyle S}$: the correspondence is given by ${\displaystyle I\mapsto f(I)}$ and the pre-image ${\displaystyle J\mapsto f^{-1}(J)}$. Moreover, for commutative rings, this bijective correspondence restricts to prime ideals, maximal ideals, and radical ideals (see the Types of ideals section for the definitions of these ideals).
• (For those who know modules) If M is a left R-module and ${\displaystyle S\subset M}$ a subset, then the annihilator ${\displaystyle \operatorname {Ann} _{R}(S)=\{r\in R\mid rs=0,s\in S\}}$ of S is a left ideal. Given ideals ${\displaystyle {\mathfrak {a}},{\mathfrak {b}}}$ of a commutative ring R, the R-annihilator of ${\displaystyle ({\mathfrak {b}}+{\mathfrak {a}})/{\mathfrak {a}}}$ is an ideal of R called the ideal quotient of ${\displaystyle {\mathfrak {a}}}$ by ${\displaystyle {\mathfrak {b}}}$ and is denoted by ${\displaystyle ({\mathfrak {a}}:{\mathfrak {b}})}$; it is an instance of idealizer in commutative algebra.
• Let ${\displaystyle {\mathfrak {a}}_{i},i\in S}$ be an ascending chain of left ideals in a ring R; i.e., ${\displaystyle S}$ is a totally ordered set and ${\displaystyle {\mathfrak {a}}_{i}\subset {\mathfrak {a}}_{j}}$ for each ${\displaystyle i. Then the union ${\displaystyle \textstyle \bigcup _{i\in S}{\mathfrak {a}}_{i}}$ 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 ${\displaystyle E\subset R}$ is a possibly empty subset and ${\displaystyle {\mathfrak {a}}_{0}\subset R}$ is a left ideal that is disjoint from E, then there is an ideal that is maximal among the ideals containing ${\displaystyle {\mathfrak {a}}_{0}}$ and disjoint from E. (Again this is still valid if the ring R lacks the unity 1.) When ${\displaystyle R\neq 0}$, taking ${\displaystyle {\mathfrak {a}}_{0}=(0)}$ and ${\displaystyle E=\{1\}}$, 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 ${\displaystyle RX}$. Such an ideal exists since it is the intersection of all left ideals containing X. Equivalently, ${\displaystyle RX}$ is the set of all the (finite) left R-linear combinations of elements of X over R:
${\displaystyle RX=\{r_{1}x_{1}+\dots +r_{n}x_{n}\mid n\in \mathbb {N} ,r_{i}\in R,x_{i}\in X\}.}$
(since such a span is the smallest left ideal 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.,
${\displaystyle RXR=\{r_{1}x_{1}s_{1}+\dots +r_{n}x_{n}s_{n}\mid n\in \mathbb {N} ,r_{i}\in R,s_{i}\in R,x_{i}\in X\}.\,}$
• 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 ${\displaystyle Rx}$ (resp. ${\displaystyle xR,RxR}$). The principal two-sided ideal ${\displaystyle RxR}$ is often also denoted by ${\displaystyle (x)}$. If ${\displaystyle X=\{x_{1},\dots ,x_{n}\}}$ is a finite set, then ${\displaystyle RXR}$ is also written as ${\displaystyle (x_{1},\dots ,x_{n})}$.
• In the ring ${\displaystyle \mathbb {Z} }$ of integers, every ideal can be generated by a single number (so ${\displaystyle \mathbb {Z} }$ is a principal ideal domain), as a consequence of Euclidean division (or some other way).
• There is a bijective correspondence between ideals and congruence relations (equivalence relations that respect the ring structure) on the ring: Given an ideal I of a ring R, let x ~ y if xyI. Then ~ is a congruence relation on R. Conversely, given a congruence relation ~ on R, let I = { x | x ~ 0 }. Then I is an ideal of R.

Types of ideals

To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles.

Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings. Different types of ideals are studied because they can be used to construct different types of factor rings.

• Maximal ideal : A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I a proper subset of J. The factor ring of a maximal ideal is a simple ring in general and is a field for commutative rings. [5]
• Minimal ideal : A nonzero ideal is called minimal if it contains no other nonzero ideal.
• Prime ideal : A proper ideal I is called a prime ideal if for any a and b in R, if ab is in I, then at least one of a and b is in I. The factor ring of a prime ideal is a prime ring in general and is an integral domain for commutative rings.
• Radical ideal or semiprime ideal: A proper ideal I is called radical or semiprime if for any a in R, if an 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 bn is in I for some natural number n. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime.
• Principal ideal : An ideal generated by one element.
• Finitely generated ideal: This type of ideal is finitely generated as a module.
• Primitive ideal : A left primitive ideal is the annihilator of a simple left module.
• Irreducible ideal : An ideal is said to be irreducible if it cannot be written as an intersection of ideals which properly contain it.
• Comaximal ideals: Two ideals ${\displaystyle {\mathfrak {i}},{\mathfrak {j}}}$ are said to be comaximal if ${\displaystyle x+y=1}$ for some ${\displaystyle x\in {\mathfrak {i}}}$ and ${\displaystyle y\in {\mathfrak {j}}}$.
• 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.

Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details:

• Fractional ideal : This is usually defined when R is a commutative domain with quotient field K. Despite their names, fractional ideals are R submodules of K with a special property. If the fractional ideal is contained entirely in R, then it is truly an ideal of R.
• Invertible ideal : Usually an invertible ideal A is defined as a fractional ideal for which there is another fractional ideal B such that AB=BA=R. Some authors may also apply "invertible ideal" to ordinary ring ideals A and B with AB=BA=R in rings other than domains.

Ideal operations

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

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

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

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

i.e. the product is the ideal generated by all products of the form ab with a in ${\displaystyle {\mathfrak {a}}}$ and b in ${\displaystyle {\mathfrak {b}}}$.

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

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

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

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

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

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

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

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

(More generally, the difference between a product and an intersection of ideals is measured by the Tor functor: ${\displaystyle \operatorname {Tor} _{1}^{R}(R/{\mathfrak {a}},R/{\mathfrak {b}})=({\mathfrak {a}}\cap {\mathfrak {b}})/{\mathfrak {a}}{\mathfrak {b}}.}$ [6] )

An integral domain is called a Dedekind domain if for each pair of ideals ${\displaystyle {\mathfrak {a}}\subset {\mathfrak {b}}}$, there is an ideal ${\displaystyle {\mathfrak {c}}}$ such that ${\displaystyle {\mathfrak {\mathfrak {a}}}={\mathfrak {b}}{\mathfrak {c}}}$. [7] 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 ${\displaystyle \mathbb {Z} }$ we have

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

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

Let ${\displaystyle R=\mathbb {C} [x,y,z,w]}$ and let ${\displaystyle I=(z,w),{\text{ }}J=(x+z,y+w),{\text{ }}K=(x+z,w)}$. Then,

• ${\displaystyle I+J=(z,w,x+z,y+w)=(x,y,z,w)}$ and ${\displaystyle I+K=(z,w,x+z)}$
• ${\displaystyle IJ=(z(x+z),z(y+w),w(x+z),w(y+w))=(z^{2}+xz,zy+wz,wx+wz,wy+w^{2})}$
• ${\displaystyle IK=(xz+z^{2},zw,xw+zw,w^{2})}$
• ${\displaystyle I\cap J=IJ}$ while ${\displaystyle I\cap K=(w,xz+z^{2})\neq IK}$

In the first computation, we see the general pattern for taking the sum of two finitely generated ideals, it is the ideal generated by the union of their generators. In the last three we observe that products and intersections agree whenever the two ideals intersect in the zero ideal. These computations can be checked using Macaulay2. [8] [9] [10]

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 ${\displaystyle J=\operatorname {Jac} (R)}$ of R is the intersection of all primitive ideals. Equivalently,

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

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

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

For a not-necessarily-commutative ring, it is a general fact that ${\displaystyle 1-yx}$ is a unit element if and only if ${\displaystyle 1-xy}$ 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 ${\displaystyle JM=M}$, then M does not admit a maximal submodule, since if there is a maximal submodule ${\displaystyle L\subsetneq M}$, ${\displaystyle J\cdot (M/L)=0}$ and so ${\displaystyle M=JM\subset L\subsetneq M}$, a contradiction. Since a nonzero finitely generated module admits a maximal submodule, in particular, one has:

If ${\displaystyle JM=M}$ and M is finitely generated, then ${\displaystyle M=0.}$

A maximal ideal is a prime ideal and so one has

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

where the intersection on the left is called the nilradical of R. As it turns out, ${\displaystyle \operatorname {nil} (R)}$ is also the set of nilpotent elements of R.

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

Extension and contraction of an ideal

Let A and B be two commutative rings, and let f : AB be a ring homomorphism. If ${\displaystyle {\mathfrak {a}}}$ is an ideal in A, then ${\displaystyle f({\mathfrak {a}})}$ need not be an ideal in B (e.g. take f to be the inclusion of the ring of integers Z into the field of rationals Q). The extension${\displaystyle {\mathfrak {a}}^{e}}$ of ${\displaystyle {\mathfrak {a}}}$ in B is defined to be the ideal in B generated by ${\displaystyle f({\mathfrak {a}})}$. Explicitly,

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

If ${\displaystyle {\mathfrak {b}}}$ is an ideal of B, then ${\displaystyle f^{-1}({\mathfrak {b}})}$ is always an ideal of A, called the contraction${\displaystyle {\mathfrak {b}}^{c}}$ of ${\displaystyle {\mathfrak {b}}}$ to A.

Assuming f : AB is a ring homomorphism, ${\displaystyle {\mathfrak {a}}}$ is an ideal in A, ${\displaystyle {\mathfrak {b}}}$ is an ideal in B, then:

• ${\displaystyle {\mathfrak {b}}}$ is prime in B${\displaystyle \Rightarrow }$${\displaystyle {\mathfrak {b}}^{c}}$ is prime in A.
• ${\displaystyle {\mathfrak {a}}^{ec}\supseteq {\mathfrak {a}}}$
• ${\displaystyle {\mathfrak {b}}^{ce}\subseteq {\mathfrak {b}}}$

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

On the other hand, if f is surjective and ${\displaystyle {\mathfrak {a}}\supseteq \ker f}$ then:

• ${\displaystyle {\mathfrak {a}}^{ec}={\mathfrak {a}}}$ and ${\displaystyle {\mathfrak {b}}^{ce}={\mathfrak {b}}}$.
• ${\displaystyle {\mathfrak {a}}}$ is a prime ideal in A${\displaystyle \Leftrightarrow }$${\displaystyle {\mathfrak {a}}^{e}}$ is a prime ideal in B.
• ${\displaystyle {\mathfrak {a}}}$ is a maximal ideal in A${\displaystyle \Leftrightarrow }$${\displaystyle {\mathfrak {a}}^{e}}$ is a maximal ideal in B.

Remark: Let K be a field extension of L, and let B and A be the 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 ${\displaystyle {\mathfrak {a}}={\mathfrak {p}}}$ of A under extension is one of the central problems of algebraic number theory.

The following is sometimes useful: [11] a prime ideal ${\displaystyle {\mathfrak {p}}}$ is a contraction of a prime ideal if and only if ${\displaystyle {\mathfrak {p}}={\mathfrak {p}}^{ec}}$. (Proof: Assuming the latter, note ${\displaystyle {\mathfrak {p}}^{e}B_{\mathfrak {p}}=B_{\mathfrak {p}}\Rightarrow {\mathfrak {p}}^{e}}$ intersects ${\displaystyle A-{\mathfrak {p}}}$, a contradiction. Now, the prime ideals of ${\displaystyle B_{\mathfrak {p}}}$ correspond to those in B that are disjoint from ${\displaystyle A-{\mathfrak {p}}}$. Hence, there is a prime ideal ${\displaystyle {\mathfrak {q}}}$ of B, disjoint from ${\displaystyle A-{\mathfrak {p}}}$, such that ${\displaystyle {\mathfrak {q}}B_{\mathfrak {p}}}$ is a maximal ideal containing ${\displaystyle {\mathfrak {p}}^{e}B_{\mathfrak {p}}}$. One then checks that ${\displaystyle {\mathfrak {q}}}$ lies over ${\displaystyle {\mathfrak {p}}}$. The converse is obvious.)

Generalisations

Ideals can be generalised to any monoid object ${\displaystyle (R,\otimes )}$, where ${\displaystyle R}$ is the object where the monoid structure has been forgotten. A left ideal of ${\displaystyle R}$ is a subobject ${\displaystyle I}$ that "absorbs multiplication from the left by elements of ${\displaystyle R}$"; that is, ${\displaystyle I}$ is a left ideal if it satisfies the following two conditions:

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

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

An ideal can also be thought of as a specific type of R-module. If we consider ${\displaystyle R}$ as a left ${\displaystyle R}$-module (by left multiplication), then a left ideal ${\displaystyle I}$ is really just a left sub-module of ${\displaystyle R}$. In other words, ${\displaystyle I}$ is a left (right) ideal of ${\displaystyle R}$ if and only if it is a left (right) ${\displaystyle R}$-module which is a subset of ${\displaystyle R}$. ${\displaystyle I}$ is a two-sided ideal if it is a sub-${\displaystyle R}$-bimodule of ${\displaystyle R}$.

Example: If we let ${\displaystyle R=\mathbb {Z} }$, an ideal of ${\displaystyle \mathbb {Z} }$ is an abelian group which is a subset of ${\displaystyle \mathbb {Z} }$, i.e. ${\displaystyle m\mathbb {Z} }$ for some ${\displaystyle m\in \mathbb {Z} }$. So these give all the ideals of ${\displaystyle \mathbb {Z} }$.

Notes

1. Some authors call the zero and unit ideals of a ring R the trivial ideals of R.
2. 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.

Related Research Articles

In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal.

In algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by , is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an affine scheme.

In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R.

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.

In mathematics, more specifically ring theory, the Jacobson radical of a ring is the ideal consisting of those elements in that annihilate all simple right -modules. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left-right symmetric. The Jacobson radical of a ring is frequently denoted by or ; the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in.

Hilbert's Nullstellensatz is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, a branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert who proved the Nullstellensatz and several other important related theorems named after him.

In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.

In commutative ring theory, a branch of mathematics, the radical of an ideal is an ideal such that an element is in the radical if and only if some power of is in . A radical ideal is an ideal that is equal to its own radical. The radical of a primary ideal is a prime ideal.

In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing one so that it consists of fractions such that the denominator s belongs to a given subset S of R. If S is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the ring Q of rational numbers from the ring Z of integers.

In mathematics, the annihilator of a subset S of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by an element of S.

In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring R, then a function is called an R-module homomorphism or an R-linear map if for any x, y in M and r in R,

In algebra, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact.

In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D.

In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals. The theorem was first proven by Emanuel Lasker (1905) for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by Emmy Noether (1921).

In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring and its finitely generated modules. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative ring behave like vector spaces over a field. It is an important tool in algebraic geometry, because it allows local data on algebraic varieties, in the form of modules over local rings, to be studied pointwise as vector spaces over the residue field of the ring.

In commutative algebra, an element b of a commutative ring B is said to be integral overA, a subring of B, if there are n ≥ 1 and aj in A such that

In commutative algebra, the support of a module M over a commutative ring A is the set of all prime ideals of A such that . It is denoted by . The support is, by definition, a subset of the spectrum of A.

In abstract algebra, an associated prime of a module M over a ring R is a type of prime ideal of R that arises as an annihilator of a (prime) submodule of M. The set of associated primes is usually denoted by and sometimes called the assassin or assassinator of M.

References

1. John Stillwell (2010). Mathematics and its history. p. 439.
2. Harold M. Edwards (1977). Fermat's last theorem. A genetic introduction to algebraic number theory. p. 76.
3. Everest G., Ward T. (2005). An introduction to number theory. p. 83.
4. Lang 2005 , Section III.2
5. Because simple commutative rings are fields. See Lam (2001). A First Course in Noncommutative Rings. p. 39.
6. Eisenbud , Exercise A 3.17
7. Milnor , page 9.
8. "ideals". www.math.uiuc.edu. Archived from the original on 2017-01-16. Retrieved 2017-01-14.
9. "sums, products, and powers of ideals". www.math.uiuc.edu. Archived from the original on 2017-01-16. Retrieved 2017-01-14.
10. "intersection of ideals". www.math.uiuc.edu. Archived from the original on 2017-01-16. Retrieved 2017-01-14.
11. Atiyah–MacDonald , Proposition 3.16.