In
ring theory, a branch of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, an ideal of a
ring is a special
subset
In mathematics, 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 of ...
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 languag ...
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 a way similar to how, in
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
, a
normal subgroup can be used to construct a
quotient group.
Among the integers, the ideals correspond one-for-one with the
non-negative integers: 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 that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
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 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
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
).
The related, but distinct, concept of an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
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 is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity.
History
Ernst Kummer invented the concept of
ideal numbers to serve as the "missing" factors in number rings in which unique factorization fails; here the word "ideal" is in the sense of existing in imagination only, in analogy with "ideal" objects in geometry such as points at infinity.
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
, let
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 of
# For every
and every
, the product
is in
.
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
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
s, 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 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
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, 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 of ...
). 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 (), 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 languag ...
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
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
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
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s ''f'' from
to
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 In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
The center of a simpl ...
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
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ...
over a skew-field is a simple ring.
* If
is a
ring homomorphism, 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
#Extension and contraction of an ideal below.
* 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 of
by
and is denoted by
; it is an instance of
idealizer 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 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
.
* 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 relations (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 rings. Different types of ideals are studied because they can be used to construct different types of factor rings.
*
Maximal ideal: A proper ideal ''I'' is called a maximal ideal if there exists no other proper ideal ''J'' with ''I'' a proper subset of ''J''. The factor ring of a maximal ideal is a
simple ring In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
The center of a simpl ...
in general and is a
field for commutative rings.
*
Minimal ideal In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring ''R'' is a nonzero right ideal which contains no other nonzero right ideal. Likewise, a minimal left ideal is a nonzero left ideal of ''R'' containing no other ...
: 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 that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
: 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
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''. 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 Johnn ...
left
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
.
*
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
for some
and
.
*
Regular ideal In mathematics, especially ring theory, a regular ideal can refer to multiple concepts.
In operator theory, a right ideal \mathfrak in a (possibly) non-unital ring ''A'' is said to be regular (or modular) if there exists an element ''e'' in ''A' ...
: This term has multiple uses. See the article for a list.
*
Nil ideal In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent., p. 194
The nilradical of a commutative ring is an example of a nil ideal; in fact, it is ...
: 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
In mathematics, a system of parameters for a local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public ...
.
Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details:
*
Fractional ideal: This is usually defined when ''R'' is a commutative domain with
quotient field ''K''. Despite their names, fractional ideals are ''R'' submodules of ''K'' with a special property. If the fractional ideal is contained entirely in ''R'', then it is truly an ideal of ''R''.
*
Invertible ideal: Usually an invertible ideal ''A'' is defined as a fractional ideal for which there is another fractional ideal ''B'' such that . 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
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
Complete may refer to:
Logic
* Completeness (logic)
* 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, which implies t ...
modular lattice
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self- dual condition,
;Modular law: implies
where are arbitrary elements in the lattice, ≤ is the partial order, and & ...
. 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 In mathematics, quantales are certain partially ordered algebraic structures that generalize locales ( point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis (C*-algebras, von Neumann ...
.
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
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to co ...
:
)
An integral domain is called a
Dedekind domain 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.
Examples of ideal operations
In
we have
:
since
is the set of integers which are divisible by both
and
.
Let