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 integer
s, such as the even numbers
or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any 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
Among the integers, the ideals correspond one-for-one with the non-negative integer
s: in this ring, every ideal is a principal ideal
consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. For instance, the prime ideal
s of a ring are analogous to prime number
s, and the Chinese remainder theorem
can be generalized to ideals. There is a version of 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.
invented the concept of ideal number
s to serve as the "missing" factors in number rings in which unique factorization fails; here the word "ideal" is in the sense of existing in imagination only, in analogy with "ideal" objects in geometry such as points at infinity.
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.
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
be its additive group
. A subset
is called a left ideal of
if it is an additive subgroup 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
# For every
, the product
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 module
s, 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 ℤ''n''
of integers modulo a given integer ''n'' ∈ ℤ (note that ℤ is a commutative ring). The key observation here is that we obtain ℤ''n''
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 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''
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
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. If ''R'' is not necessarily commutative, the above construction still works using two-sided ideals.
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
). Note: a left ideal
is proper if and only if it does not contain a unit element, since if
is a unit element, then
. Typically there are plenty of proper ideals. In fact, if ''R'' is a skew-field
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
for some nonzero
for some nonzero
* The even integer
s form an ideal in the ring
of all integers; it is usually denoted by
. 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
* The set of all polynomial
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
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
of all continuous function
s ''f'' from
under pointwise multiplication
contains the ideal of all continuous functions ''f'' such that ''f''(1) = 0. Another ideal in
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
. 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.
is a ring homomorphism
, then the kernel
is a two-sided ideal of
. By definition,
, and thus if
is not the zero ring (so
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 #Extension and contraction of an ideal
* 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).
* (For those who know modules) 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
and is denoted by
; it is an instance of idealizer
in commutative algebra.
be an ascending chain
of left ideals in a ring ''R''; i.e.,
is a totally ordered set and
. 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
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
, 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
*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
). The principal two-sided ideal
is often also denoted by
is a finite set, then
is also written as
* In the ring
of integers, every ideal can be generated by a single number (so
is a principal ideal domain
), as a consequence of Euclidean division
(or some other way).
*There is a bijective correspondence between ideals and congruence relation
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
s. 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.
* 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 ''a''''n''
is in ''I'' for some ''n'', then ''a'' is in ''I''. The factor ring of a radical ideal is a semiprime ring
for general rings, and is a reduced ring
for commutative rings.
* Primary ideal
: An ideal ''I'' is called a primary ideal if for all ''a'' and ''b'' in ''R'', if ''ab'' is in ''I'', then at least one of ''a'' and ''b''''n''
is in ''I'' for some natural number
''n''. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime.
* Principal ideal
: An ideal generated by ''one'' element.
* 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
* 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
are said to be comaximal if
* 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:
: 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''.
: 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.
The sum and product of ideals are defined as follows. For
, left (resp. right) ideals of a ring ''R'', their sum is
which is a left (resp. right) ideal,
i.e. the product is the ideal generated by all products of the form ''ab'' with ''a'' in
and ''b'' in
is the smallest left (resp. right) ideal containing both
(or the union
), while the product
is contained in the intersection of
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
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
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
if for each pair of ideals
, there is an ideal
. 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
is the set of integers which are divisible by both