In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and more specifically in
ring theory, an ideal of a
ring is a special
subset
In mathematics, a 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 a ...
of its elements. Ideals generalize certain subsets of the
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 in linear algebra. ...
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 conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
can be used to construct a
quotient group
A quotient group or factor group is a mathematical 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 "factored out"). For ex ...
.
Among the integers, the ideals correspond one-for-one with the
non-negative integer
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
s: in this ring, every ideal is a
principal ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...
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 th ...
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 (mathematics), 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 ...
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 mathematics, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily un ...
(a type of ring important in
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
).
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 intr ...
is derived from the notion of ideal in ring theory. A
fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral do ...
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. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of h ...
invented the concept of
ideal number
In number theory, an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the r ...
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 who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. H ...
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. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In Mathematical analysis, analysis, h ...
's book ''
Vorlesungen über Zahlentheorie
(; German for ''Lectures on Number Theory'') is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Others were written by Leopold K ...
'', 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 and philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad range of fundamental idea ...
and especially
Emmy Noether
Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
.
Definitions
Given a
ring , a left ideal is a subset of that is a
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation  ...
of the
additive group
An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.
This terminology is widely used with structu ...
of
that "absorbs multiplication from the left by elements of "; that is,
is a left ideal if it satisfies the following two conditions:
#
is a
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation  ...
of ,
# For every
and every , the product
is in .
In other words, a left ideal is a left
submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since t ...
of , considered as a
left module over itself.
A right ideal is defined similarly, with the condition
replaced by . A two-sided ideal is a left ideal that is also a right ideal.
If the ring is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
, the three definitions are the same, and one talks simply of an ideal. In the non-commutative case, "ideal" is often used instead of "two-sided ideal".
If is a left, right or two-sided ideal, the relation
if and only if
:
is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...
on , and the set of
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es forms a left, right or bi module denoted
and called the ''
quotient
In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
'' of by . (It is an instance of a
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 ...
and is a generalization of
modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mo ...
.)
If the ideal is two-sided,
is a ring, and the function
:
that associates to each element of its equivalence class is a
surjective
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
ring homomorphism
In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
that has the ideal as its
kernel. Conversely, the kernel of a ring homomorphism is a two-sided ideal. Therefore, ''the two-sided ideals are exactly the kernels of ring homomorphisms.''
Note on convention
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. For a rng , a left ideal is a with the additional property that
is in for every
and every
. (Right and two-sided ideals are defined similarly.) For a ring, an ideal (say a left ideal) is rarely a subring; since a subring shares the same multiplicative identity with the ambient ring , if were a subring, for every
, we have
i.e.,
.
The notion of an ideal does not involve associativity; thus, an ideal is also defined for
non-associative ring
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if ...
s (often without the multiplicative identity) such as a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
.
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
since it is precisely the two-sided ideal generated (see below) by the unity . Also, the set
consisting of only the additive identity 0
''R'' forms a two-sided ideal called the zero ideal and is denoted by .
[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, a set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset ...
). Note: a left ideal
is proper if and only if it does not contain a unit element, since if
is a unit element, then
for every . Typically there are plenty of proper ideals. In fact, if ''R'' is a
skew-field, then
are its only ideals and conversely: that is, a nonzero ring ''R'' is a skew-field if
are the only left (or right) ideals. (Proof: if
is a nonzero element, then the principal left ideal
(see below) is nonzero and thus
; i.e.,
for some nonzero . Likewise,
for some nonzero
. Then
.)
* The even
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s form an ideal in the ring
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 . More generally, the set of all integers divisible by a fixed integer
is an ideal denoted . In fact, every non-zero ideal of the ring
is generated by its smallest positive element, as a consequence of
Euclidean division
In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
, so
is a
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some author ...
.
* The set of all
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s with real coefficients that are divisible by the polynomial
is an ideal in the ring of all real-coefficient polynomials .
* Take a ring
and positive integer . For each , the set of all
matrices
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the ...
with entries in
whose
-th row is zero is a right ideal in the ring
of all
matrices with entries in . It is not a left ideal. Similarly, for each , the set of all
matrices whose
-th ''column'' is zero is a left ideal but not a right ideal.
* The ring
of all
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s
from
to
under
pointwise multiplication contains the ideal of all continuous functions
such that . Another ideal in
is given by those functions that vanish for large enough arguments, i.e. those continuous functions
for which there exists a number
such that
whenever .
* A ring is called a
simple ring if it is nonzero and has no two-sided ideals other than . 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
is a
ring homomorphism
In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
, then the kernel
is a two-sided ideal of . By definition, , and thus if
is not the zero ring (so ), then
is a proper ideal. More generally, for each left ideal ''I'' of ''S'', the pre-image
is a left ideal. If ''I'' is a left ideal of ''R'', then
is a left ideal of the subring
of ''S'': unless ''f'' is surjective,
need not be an ideal of ''S''; see also .
* Ideal correspondence: Given a surjective ring homomorphism , there is a bijective order-preserving correspondence between the left (resp. right, two-sided) ideals of
containing the kernel of
and the left (resp. right, two-sided) ideals of
: the correspondence is given by
and the pre-image . Moreover, for commutative rings, this bijective correspondence restricts to prime ideals, maximal ideals, and radical ideals (see the
Types of ideals section for the definitions of these ideals).
* If ''M'' is a left ''R''-
module and
a subset, then the
annihilator of ''S'' is a left ideal. Given ideals
of a commutative ring ''R'', the ''R''-annihilator of
is an ideal of ''R'' called the
ideal quotient of
by
and is denoted by ; it is an instance of
idealizer In abstract algebra, the idealizer of a subsemigroup ''T'' of a semigroup ''S'' is the largest subsemigroup of ''S'' in which ''T'' is an Semigroup#Subsemigroups and ideals, ideal. Such an idealizer is given by
:\mathbb_S(T)=\.
In ring theory, if ...
in commutative algebra.
* Let
be an
ascending chain of left ideals in a ring ''R''; i.e.,
is a totally ordered set and
for each . Then the union
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 (that is, every totally ordered subset) necessarily contains at least on ...
proves the following: if
is a possibly empty subset and
is a left ideal that is disjoint from ''E'', then there is an ideal that is maximal among the ideals containing
and disjoint from ''E''. (Again this is still valid if the ring ''R'' lacks the unity 1.) When
, taking
and , 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 . Such an ideal exists since it is the intersection of all left ideals containing ''X''. Equivalently,
is the set of all the
(finite) left ''R''-linear combinations of elements of ''X'' over ''R'':
(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.,
* 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
(resp. ). The principal two-sided ideal
is often also denoted by . If
is a finite set, then
is also written as .
* 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
of a ring , let
if . Then
is a congruence relation on . Conversely, given a congruence relation
on , let . Then
is an ideal of .
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, 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 ...
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 that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
: A proper ideal is called a maximal ideal if there exists no other proper ideal with a proper subset of . 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.
* Zero ideal: the ideal
.
* Unit ideal: the whole ring (being the ideal generated by
).
*
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 th ...
: A proper ideal
is called a prime ideal if for any
and
in , if
is in , then at least one of
and
is in . The factor ring of a prime ideal is a
prime ring in general and is an
integral domain
In mathematics, 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 divisibilit ...
for commutative rings.
*
Radical ideal or
semiprime ideal: A proper ideal is called radical or semiprime if for any in
, if is in for some , then is in . The factor ring of a radical ideal is a
semiprime ring for general rings, and is a
reduced ring for commutative rings.
*
Primary ideal
In mathematics, specifically commutative algebra, a proper ideal ''Q'' of a commutative ring ''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y'n'' is also an element of ''Q'', for some ''n'' > 0. ...
: An ideal is called a primary ideal if for all and in , if is in , then at least one of and is in for some
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime.
*
Principal ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...
: 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
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by John ...
left
module.
*
Irreducible ideal In mathematics, a proper ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two strictly larger ideals..
Examples
* Every prime ideal is irreducible. Let J and K be ideals of a commutative ring R ...
: 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 , are said to be comaximal if
for some
and .
*
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 In mathematics, more specifically ring theory, an ideal ''I'' of a ring ''R'' is said to be a nilpotent ideal if there exists a natural number ''k'' such that ''I'k'' = 0. By ''I'k'', it is meant the additive subgroup generated by the set o ...
: Some power of it is zero.
*
Parameter ideal: an ideal generated by a
system of parameters
In mathematics, a system of parameters for a local ring, local Noetherian ring of Krull dimension ''d'' with maximal ideal ''m'' is a set of elements ''x''1, ..., ''x'd'' that satisfies any of the following equivalent conditions:
# ''m'' is a M ...
.
*
Perfect ideal
In commutative algebra, a perfect ideal is a proper ideal I in a Noetherian ring R such that its grade equals the projective dimension of the associated quotient ring.
\textrm(I)=\textrm\dim(R/I).
A perfect ideal is unmixed.
For a regular ...
: A proper ideal in a Noetherian ring
is called a perfect ideal if its
grade equals the
projective dimension
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various equivalent characterizatio ...
of the associated quotient ring,
. A perfect ideal is
unmixed.
*
Unmixed ideal: A proper ideal in a Noetherian ring
is called an unmixed ideal (in height) if the height of is equal to the height of every
associated prime
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 \operatorname_R(M) ...
of
. (This is stronger than saying that
is
equidimensional. See also
equidimensional ring.
Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details:
*
Fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral do ...
: This is usually defined when
is a commutative domain with
quotient field . Despite their names, fractional ideals are not necessarily ideals. A fractional ideal of
is an
-submodule of
for which there exists a non-zero
such that
. If the fractional ideal is contained entirely in
, then it is truly an ideal of
.
*
Invertible ideal: Usually an invertible ideal is defined as a fractional ideal for which there is another fractional ideal such that . Some authors may also apply "invertible ideal" to ordinary ring ideals and with in rings other than domains.
Ideal operations
The sum and product of ideals are defined as follows. For
and , left (resp. right) ideals of a ring ''R'', their sum is
:
,
which is a left (resp. right) ideal,
and, if
are two-sided,
:
i.e. the product is the ideal generated by all products of the form ''ab'' with ''a'' in
and ''b'' in .
Note
is the smallest left (resp. right) ideal containing both
and
(or the union ), while the product
is contained in the intersection of
and .
The distributive law holds for two-sided ideals ,
* ,
* .
If a product is replaced by an intersection, a partial distributive law holds:
:
where the equality holds if
contains
or
.
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
In mathematics, a distributive lattice is a lattice (order), lattice in which the operations of join and meet distributivity, distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice o ...
. The three operations of intersection, sum (or join), and product make the set of ideals of a commutative ring into a
quantale.
If
are ideals of a commutative ring ''R'', then
in the following two cases (at least)
*
*
is generated by elements that form a
regular sequence modulo .
(More generally, the difference between a product and an intersection of ideals is measured by the
Tor functor: .)
An integral domain is called a
Dedekind domain
In mathematics, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily un ...
if for each pair of ideals
, there is an ideal
such that . 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 is prime or can be represented uniquely as a product of prime numbers, ...
.
Examples of ideal operations
In
we have
:
since
is the set of integers that are divisible by both
and .
Let