ideal (ring theory)
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In
ring theory In algebra, ring theory is the study of ring (mathematics), 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 ...
, a branch of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, an ideal of a ring is a special
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of its elements. Ideals generalize certain subsets of the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
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 and absorption properties are the defining properties of an ideal. An ideal can be used to construct a
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space (linear algebra), quo ...
in a way similar to how, in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under inner automorphism, conjugation by members of the Group (mathematics), group of which it is a part. In o ...
can be used to construct a
quotient group A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factor ...
. Among the integers, the ideals correspond one-for-one with the
non-negative integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s: in this ring, every ideal is a
principal ideal In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
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 In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic_properties, integers. The prime ideals for the integers are the sets that contain all the ...
s of a ring are analogous to
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...
s, and the
Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
can be generalized to ideals. There is a version of unique prime factorization for the ideals of a
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every Ideal (ring_theory)#Examples and properties, nonzero proper ideal factors into a product of prime ideals. It can be shown t ...
(a type of ring important in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...
). The related, but distinct, concept of an ideal in
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...
is derived from the notion of ideal in ring theory. A
fractional ideal In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity.


History

Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned ...
invented the concept of
ideal number In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or ...
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. In 1876,
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathemati ...
replaced Kummer's undefined concept by concrete sets of numbers, sets that he called ideals, in the third edition of
Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathem ...
's book ''
Vorlesungen über Zahlentheorie (German for ''Lectures on Number Theory'') is the name of several different textbooks of number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the i ...
'', to which Dedekind had added many supplements. Later the notion was extended beyond number rings to the setting of polynomial rings and other commutative rings by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
and especially
Emmy Noether Amalie Emmy NoetherEmmy (given name), Emmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promot ...
.


Definitions and motivation

For an arbitrary ring (R,+,\cdot), let (R,+) be its
additive group An additive group is a group (mathematics), group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian group, abelian, and typically written using the symbol + for its binary operation. This termin ...
. A subset I is called a left ideal of R if it is an additive subgroup of R that "absorbs multiplication from the left by elements of R"; that is, I is a left ideal if it satisfies the following two conditions: # (I,+) is a
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ...
of (R,+), # For every r \in R and every x \in I, the product r x is in I. A right ideal is defined with the condition replaced by . A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. In the language of modules, the definitions mean that a left (resp. right, two-sided) ideal of ''R'' is an ''R''- submodule of ''R'' when ''R'' is viewed as a left (resp. right, bi-) ''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 ℤ/''n''ℤ of integers modulo ''n'' given integer (note that ℤ is a commutative ring). The key observation here is that we obtain ℤ/''n''ℤ by taking the integer line ℤ and wrapping it around itself so that various integers get identified. In doing so, we must satisfy 2 requirements: 1) ''n'' must be identified with 0 since ''n'' is congruent to 0 modulo ''n.'' 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 of all integers congruent to 0 modulo ''n''. That is, we must wrap ℤ around itself infinitely many times so that the integers ..., , , , , ... will all align with 0. If we look at what properties this set must satisfy in order to ensure that ℤ/''n''ℤ is a ring, then we arrive at the definition of an ideal. Indeed, one can directly verify that ''n''ℤ is an ideal of ℤ. Remark. Identifications with elements other than 0 also need to be made. For example, the elements in must be identified with 1, the elements in must be identified with 2, and so on. Those, however, are uniquely determined by ''n''ℤ since ℤ is an additive group. We can make a similar construction in any commutative ring ''R'': start with an arbitrary , and then identify with 0 all elements of the ideal 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 , and then identify with 0 all the elements in the ideal generated by ''S'': the smallest ideal (''S'') such that . 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 , 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 . 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 In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division (mathematics), division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to a ...
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'': # ''I'' is an additive subgroup of ''R'': the zero 0 of ''R'' is a "zero" , and if and are "zeros", then is a "zero" too. # Any multiplied by a "zero" is a "zero" . 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 (1) since it is precisely the two-sided ideal generated (see below) by the unity 1_R. Also, the set \ consisting of only the additive identity 0''R'' forms a two-sided ideal called the zero ideal and is denoted by (0).Some authors call the zero and unit ideals of a ring ''R'' the trivial ideals of ''R''. 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 In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
). Note: a left ideal \mathfrak is proper if and only if it does not contain a unit element, since if u \in \mathfrak is a unit element, then r = (r u^) u \in \mathfrak for every r \in R. 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 y. Likewise, zy = 1 for some nonzero z. Then z = z(yx) = (zy)x = x.) * The even
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
s form an ideal in the ring \mathbb of all integers; it is usually denoted by 2\mathbb. 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 n\mathbb. * The set of all
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s with real coefficients which are divisible by the polynomial ''x''2 + 1 is an ideal in the ring of all polynomials. * The set of all ''n''-by-''n''
matrices Matrix most commonly refers to: * The Matrix (franchise), ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within Th ...
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 C(\mathbb) of all
continuous function In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value ...
s ''f'' from \mathbb to \mathbb under pointwise multiplication contains the ideal of all continuous functions ''f'' such that ''f''(1) = 0. Another ideal in C(\mathbb) 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 > ''L''. * A ring is called a simple ring if it is nonzero and has no two-sided ideals other than (0), (1). Thus, a skew-field is simple and a simple commutative ring is a field. The
matrix ring In abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics) ...
over a skew-field is a simple ring. * If f: R \to S is a
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function ...
, then the kernel \ker(f) = f^(0_S) is a two-sided ideal of R. By definition, f(1_R) = 1_S, and thus if S is not the zero ring (so 1_S\ne0_S), then \ker(f) is a proper ideal. More generally, for each left ideal ''I'' of ''S'', the pre-image f^(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 below. * Ideal correspondence: Given a surjective ring homomorphism f: R \to S, there is a bijective order-preserving correspondence between the left (resp. right, two-sided) ideals of R 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 J \mapsto f^(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 S \subset M a subset, then the annihilator \operatorname_R(S) = \ of ''S'' is a left ideal. Given ideals \mathfrak, \mathfrak of a commutative ring ''R'', the ''R''-annihilator of (\mathfrak + \mathfrak)/\mathfrak is an ideal of ''R'' called the ideal quotient of \mathfrak by \mathfrak and is denoted by (\mathfrak : \mathfrak); it is an instance of idealizer in commutative algebra. * Let \mathfrak_i, i \in S be an ascending chain of left ideals in a ring ''R''; i.e., S is a totally ordered set and \mathfrak_i \subset \mathfrak_j for each i < j. Then the union \textstyle \bigcup_ \mathfrak_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 Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (order theory), chain (that is, every total order, totally ordered subset) n ...
proves the following: if E \subset R is a possibly empty subset and \mathfrak_0 \subset R is a left ideal that is disjoint from ''E'', then there is an ideal that is maximal among the ideals containing \mathfrak_0 and disjoint from ''E''. (Again this is still valid if the ring ''R'' lacks the unity 1.) When R \ne 0, taking \mathfrak_0 = (0) and E = \, 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 In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a zero ring, nonzero ring (mathematics), ring has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfin ...
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 RX. Such an ideal exists since it is the intersection of all left ideals containing ''X''. Equivalently, RX is the set of all the (finite) left ''R''-linear combinations of elements of ''X'' over ''R'': *:RX = \. :(since such a span is the smallest left ideal containing ''X''.)If ''R'' does not have a unit, then the internal descriptions above must be modified slightly. In addition to the finite sums of products of things in ''X'' with things in ''R'', we must allow the addition of ''n''-fold sums of the form , and ''n''-fold sums of the form for every ''x'' in ''X'' and every ''n'' in the natural numbers. When ''R'' has a unit, this extra requirement becomes superfluous. A right (resp. two-sided) ideal generated by ''X'' is defined in the similar way. For "two-sided", one has to use linear combinations from both sides; i.e., ::RXR = \.\, *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 Rx (resp. xR, RxR). The principal two-sided ideal RxR is often also denoted by (x). If X = \ is a finite set, then RXR is also written as (x_1, \dots, x_n). * In the ring \mathbb of integers, every ideal can be generated by a single number (so \mathbb is a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal (ring theory), ideal is principal ideal, principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commuta ...
), as a consequence of
Euclidean division In arithmetic Arithmetic () is an elementary part of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantitie ...
(or some other way). *There is a bijective correspondence between ideals and
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group (mathematics), group, ring (mathematics), ring, or vector space) that is compatible with the structure in the ...
s (equivalence relations that respect the ring structure) on the ring: Given an ideal ''I'' of a ring ''R'', let if . Then ~ is a congruence relation on ''R''. Conversely, given a congruence relation ~ on ''R'', let . 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 ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory stu ...
s. Different types of ideals are studied because they can be used to construct different types of factor rings. *
Maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal (ring theory), ideal that is maximal element, maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring (m ...
: 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. * Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal. *
Prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic_properties, integers. The prime ideals for the integers are the sets that contain all the ...
: 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 abstract algebra, a zero ring, nonzero ring (mathematics), ring ''R'' is a prime ring if for any two elements ''a'' and ''b'' of ''R'', ''arb'' = 0 for all ''r'' in ''R'' implies that either ''a'' = 0 or ''b'' = 0. This definition can be regarde ...
in general and is an
integral domain In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
for commutative rings. *
Radical ideal In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studi ...
or semiprime ideal: A proper ideal ''I'' is called radical or semiprime if for any ''a'' in ''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 In ring theory, a branch of mathematics, a ring (mathematics), ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, ''x''2 = 0 im ...
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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
''n''. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime. *
Principal ideal In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
: 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 In mathematics, a ideal_(ring_theory)#Examples_and_properties, proper ideal (ring theory), ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection (set theory), intersection of two strictly larger ideals.. ...
: 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 \mathfrak, \mathfrak are said to be comaximal if x + y = 1 for some x \in \mathfrak and y \in \mathfrak. * Regular ideal: This term has multiple uses. See the article for a list. *
Nil ideal In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...
: An ideal is a nil ideal if each of its elements is nilpotent. *
Nilpotent ideal In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...
: 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
: This is usually defined when ''R'' is a commutative domain with
quotient field In abstract algebra, the field of fractions of an integral domain is the smallest field (mathematics), field in which it can be Embedding, embedded. The construction of the field of fractions is modeled on the relationship between the integral do ...
''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 . Some authors may also apply "invertible ideal" to ordinary ring ideals ''A'' and ''B'' with in rings other than domains.


Ideal operations

The sum and product of ideals are defined as follows. For \mathfrak and \mathfrak, left (resp. right) ideals of a ring ''R'', their sum is :\mathfrak+\mathfrak:=\, which is a left (resp. right) ideal, and, if \mathfrak, \mathfrak are two-sided, :\mathfrak \mathfrak:=\, i.e. the product is the ideal generated by all products of the form ''ab'' with ''a'' in \mathfrak and ''b'' in \mathfrak. Note \mathfrak + \mathfrak is the smallest left (resp. right) ideal containing both \mathfrak and \mathfrak (or the union \mathfrak \cup \mathfrak), while the product \mathfrak\mathfrak is contained in the intersection of \mathfrak and \mathfrak. The distributive law holds for two-sided ideals \mathfrak, \mathfrak, \mathfrak, *\mathfrak(\mathfrak + \mathfrak) = \mathfrak \mathfrak + \mathfrak \mathfrak, *(\mathfrak + \mathfrak) \mathfrak = \mathfrak\mathfrak + \mathfrak\mathfrak. If a product is replaced by an intersection, a partial distributive law holds: :\mathfrak \cap (\mathfrak + \mathfrak) \supset \mathfrak \cap \mathfrak + \mathfrak \cap \mathfrak where the equality holds if \mathfrak contains \mathfrak or \mathfrak. 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 Complete may refer to: Logic * Completeness (logic) * Complete theory, Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, ...
modular lattice. The lattice is not, in general, a
distributive lattice In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
. The three operations of intersection, sum (or join), and product make the set of ideals of a commutative ring into a
quantale In mathematics, quantales are certain partially ordered set, partially ordered algebraic structures that generalize locales (pointless topology, point free topologies) as well as various multiplicative lattice (order), lattices of Ideal (ring theory ...
. If \mathfrak, \mathfrak are ideals of a commutative ring ''R'', then \mathfrak \cap \mathfrak = \mathfrak \mathfrak in the following two cases (at least) *\mathfrak + \mathfrak = (1) *\mathfrak is generated by elements that form a regular sequence modulo \mathfrak. (More generally, the difference between a product and an intersection of ideals is measured by the
Tor functor In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
: \operatorname^R_1(R/\mathfrak, R/\mathfrak) = (\mathfrak \cap \mathfrak)/ \mathfrak \mathfrak.) An integral domain is called a
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every Ideal (ring_theory)#Examples and properties, nonzero proper ideal factors into a product of prime ideals. It can be shown t ...
if for each pair of ideals \mathfrak \subset \mathfrak, there is an ideal \mathfrak such that \mathfrak \mathfrak = \mathfrak \mathfrak. 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 In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
.


Examples of ideal operations

In \mathbb we have :(n)\cap(m) = \operatorname(n,m)\mathbb since (n)\cap(m) is the set of integers which are divisible by both n and m. Let R = \mathbb ,y,z,w/math> and let \mathfrak = (z, w), \mathfrak = (x+z,y+w),\mathfrak = (x+z, w). Then, * \mathfrak + \mathfrak = (z,w, x+z, y+w) = (x, y, z, w) and \mathfrak + \mathfrak = (z, w, x + z) * \mathfrak\mathfrak = (z(x + z), z(y + w), w(x + z), w(y + w))= (z^2 + xz, zy + wz, wx + wz, wy + w^2) * \mathfrak\mathfrak = (xz + z^2, zw, xw + zw, w^2) * \mathfrak \cap \mathfrak = \mathfrak\mathfrak while \mathfrak \cap \mathfrak = (w, xz + z^2) \neq \mathfrak\mathfrak 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.


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 In mathematics, more specifically ring theory, the Jacobson radical of a Ring (mathematics), ring R is the ideal (ring theory), ideal consisting of those elements in R that annihilator (ring theory), annihilate all Simple module, simple right R-modu ...
J = \operatorname(R) of ''R'' is the intersection of all primitive ideals. Equivalently, :J = \bigcap_ \mathfrak. Indeed, if M is a simple module and ''x'' is a nonzero element in ''M'', then Rx = M and R/\operatorname(M) = R/\operatorname(x) \simeq M, meaning \operatorname(M) is a maximal ideal. Conversely, if \mathfrak is a maximal ideal, then \mathfrak is the annihilator of the simple ''R''-module R/\mathfrak. There is also another characterization (the proof is not hard): :J = \. For a not-necessarily-commutative ring, it is a general fact that 1 - yx is a unit element if and only if 1 - xy is (see the link) and so this last characterization shows that the radical can be defined both in terms of left and right primitive ideals. The following simple but important fact (
Nakayama's lemma 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 (mathematics), ring (typically a co ...
) is built-in to the definition of a Jacobson radical: if ''M'' is a module such that JM = M, then ''M'' does not admit a maximal submodule, since if there is a maximal submodule L \subsetneq M, J \cdot (M/L) = 0 and so M = JM \subset L \subsetneq M, a contradiction. Since a nonzero
finitely generated module In mathematics, a finitely generated module is a module (mathematics), module that has a Finite set, finite generating set. A finitely generated module over a Ring (mathematics), ring ''R'' may also be called a finite ''R''-module, finite over ''R' ...
admits a maximal submodule, in particular, one has: :If JM = M and ''M'' is finitely generated, then M = 0. A maximal ideal is a prime ideal and so one has :\operatorname(R) = \bigcap_ \mathfrak \subset \operatorname(R) where the intersection on the left is called the nilradical of ''R''. As it turns out, \operatorname(R) is also the set of
nilpotent element In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
s of ''R''. If ''R'' is an
Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring (mathematics), ring that satisfies the descending chain condition on (one-sided) ideal (ring theory), ideals; that is, there is no infinite descending s ...
, then \operatorname(R) is nilpotent and \operatorname(R) = \operatorname(R). (Proof: first note the DCC implies J^n = J^ for some ''n''. If (DCC) \mathfrak \supsetneq \operatorname(J^n) is an ideal properly minimal over the latter, then J \cdot (\mathfrak/\operatorname(J^n)) = 0. That is, J^n \mathfrak = J^ \mathfrak = 0, a contradiction.)


Extension and contraction of an ideal

Let ''A'' and ''B'' be two
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring proper ...
s, and let ''f'' : ''A'' → ''B'' be a
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function ...
. If \mathfrak is an ideal in ''A'', then f(\mathfrak) 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 \mathfrak^e of \mathfrak in ''B'' is defined to be the ideal in ''B'' generated by f(\mathfrak). Explicitly, :\mathfrak^e = \Big\ If \mathfrak is an ideal of ''B'', then f^(\mathfrak) is always an ideal of ''A'', called the contraction \mathfrak^c of \mathfrak to ''A''. Assuming ''f'' : ''A'' → ''B'' is a ring homomorphism, \mathfrak is an ideal in ''A'', \mathfrak is an ideal in ''B'', then: * \mathfrak is prime in ''B'' \Rightarrow \mathfrak^c is prime in ''A''. * \mathfrak^ \supseteq \mathfrak * \mathfrak^ \subseteq \mathfrak It is false, in general, that \mathfrak being prime (or maximal) in ''A'' implies that \mathfrak^e is prime (or maximal) in ''B''. Many classic examples of this stem from algebraic number theory. For example,
embedding In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
\mathbb \to \mathbb\left\lbrack i \right\rbrack. In B = \mathbb\left\lbrack i \right\rbrack, the element 2 factors as 2 = (1 + i)(1 - i) where (one can show) neither of 1 + i, 1 - i are units in ''B''. So (2)^e is not prime in ''B'' (and therefore not maximal, as well). Indeed, (1 \pm i)^2 = \pm 2i shows that (1 + i) = ((1 - i) - (1 - i)^2), (1 - i) = ((1 + i) - (1 + i)^2), and therefore (2)^e = (1 + i)^2. On the other hand, if ''f'' is
surjective In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
and \mathfrak \supseteq \ker f then: * \mathfrak^=\mathfrak and \mathfrak^=\mathfrak. * \mathfrak is a
prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic_properties, integers. The prime ideals for the integers are the sets that contain all the ...
in ''A'' \Leftrightarrow \mathfrak^e is a prime ideal in ''B''. * \mathfrak is a
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal (ring theory), ideal that is maximal element, maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring (m ...
in ''A'' \Leftrightarrow \mathfrak^e is a maximal ideal in ''B''. Remark: Let ''K'' be a
field extension In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
of ''L'', and let ''B'' and ''A'' be the rings of integers of ''K'' and ''L'', respectively. Then ''B'' is an
integral extension In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' is ...
of ''A'', and we let ''f'' be the
inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function (mathematics), function ι, \iota that sends each element x of A to x, treated as an element of B: \iota : A\ ...
from ''A'' to ''B''. The behaviour of a
prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic_properties, integers. The prime ideals for the integers are the sets that contain all the ...
\mathfrak = \mathfrak of ''A'' under extension is one of the central problems of
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
. The following is sometimes useful: a prime ideal \mathfrak is a contraction of a prime ideal if and only if \mathfrak = \mathfrak^. (Proof: Assuming the latter, note \mathfrak^e B_ = B_ \Rightarrow \mathfrak^e intersects A - \mathfrak, a contradiction. Now, the prime ideals of B_ correspond to those in ''B'' that are disjoint from A - \mathfrak. Hence, there is a prime ideal \mathfrak of ''B'', disjoint from A - \mathfrak, such that \mathfrak B_ is a maximal ideal containing \mathfrak^e B_. One then checks that \mathfrak lies over \mathfrak. The converse is obvious.)


Generalizations

Ideals can be generalized to any monoid object (R,\otimes), where R is the object where the
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ar ...
structure has been forgotten. A left ideal of R is a
subobject In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topolo ...
I that "absorbs multiplication from the left by elements of R"; that is, I is a left ideal if it satisfies the following two conditions: # I is a
subobject In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topolo ...
of R # For every r \in (R,\otimes) and every x \in (I, \otimes), the product r \otimes x is in (I, \otimes). A right ideal is defined with the condition "r \otimes x \in (I, \otimes)" replaced by "'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 R is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone. An ideal can also be thought of as a specific type of -module. If we consider R as a left R-module (by left multiplication), then a left ideal I is really just a left sub-module of R. In other words, I is a left (right) ideal of R if and only if it is a left (right) R-module which is a subset of R. I is a two-sided ideal if it is a sub-R-bimodule of R. Example: If we let R=\mathbb, an ideal of \mathbb is an abelian group which is a subset of \mathbb, i.e. m\mathbb for some m\in\mathbb. So these give all the ideals of \mathbb.


See also

*
Modular arithmetic In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
* Noether isomorphism theorem * Boolean prime ideal theorem * Ideal theory * Ideal (order theory) *
Ideal norm In commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry an ...
* Splitting of prime ideals in Galois extensions * Ideal sheaf


Notes


References

* Atiyah, M. F. and Macdonald, I. G., '' Introduction to Commutative Algebra'', Perseus Books, 1969, * *
Michiel Hazewinkel Michiel Hazewinkel (born 22 June 1943) is a Dutch mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, da ...
, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. ''Algebras, rings and modules''. Volume 1. 2004. Springer, 2004. *


External links

* {{Authority control Algebraic structures Commutative algebra Algebraic number theory