In
ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, a branch of
mathematics, the conductor is a measurement of how far apart a
commutative ring and an
extension ring are. Most often, the larger
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
is a
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
integrally closed in its
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
, and then the conductor measures the failure of the smaller ring to be integrally closed.
The conductor is of great importance in the study of non-maximal
orders
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
in the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...
of an
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
. One interpretation of the conductor is that it measures the failure of unique factorization into
prime ideals.
Definition
Let ''A'' and ''B'' be commutative rings, and assume . The conductor
of ''A'' in ''B'' is the
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 considere ...
:
Here is viewed as a
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of ''A''-
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
, and denotes the
annihilator. More concretely, the conductor is the set
:
Because the conductor is defined as an annihilator, it is an ideal of ''A''.
If ''B'' is an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
, then the conductor may be rewritten as
:
where
is considered as a subset of the fraction field of ''B''. That is, if ''a'' is non-zero and in the conductor, then every element of ''B'' may be written as a fraction whose numerator is in ''A'' and whose denominator is ''a''. Therefore the non-zero elements of the conductor are those that suffice as common denominators when writing elements of ''B'' as quotients of elements of ''A''.
Suppose ''R'' is a ring containing ''B''. For example, ''R'' might equal ''B'', or ''B'' might be a domain and ''R'' its field of fractions. Then, because , the conductor is also equal to
:
Elementary properties
The conductor is the whole ring ''A''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
it contains and, therefore, if and only if . Otherwise, the conductor is a
proper ideal
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers pre ...
of ''A''.
If the index is finite, then , so
. In this case, the conductor is non-zero. This applies in particular when ''B'' is the ring of integers in an algebraic number field and ''A'' is an order (a subring for which is finite).
The conductor is also an ideal of ''B'', because, for any in and any in
, . In fact, an ideal ''J'' of ''B'' is contained in ''A'' if and only if ''J'' is contained in the conductor. Indeed, for such a ''J'', , so by definition ''J'' is contained in
.
Conversely
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposit ...
, the conductor is an ideal of ''A'', so any ideal contained in it is contained in ''A''. This fact implies that
is the largest ideal of ''A'' which is also an ideal of ''B''. (It can happen that there are ideals of ''A'' contained in the conductor which are not ideals of ''B''.)
Suppose that ''S'' is a
multiplicative subset In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold:
* 1 \in S,
* xy \in S for all x, y \in S.
In other words, ''S'' is closed under taking finite ...
of ''A''. Then
:
with equality in the case that ''B'' is a
finitely generated ''A''-module.
Conductors of Dedekind domains
Some of the most important applications of the conductor arise when ''B'' is a
Dedekind domain
In abstract algebra, 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 necessari ...
and is finite. For example, ''B'' can be the ring of integers of a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
and ''A'' a non-maximal order. Or, ''B'' can be the affine coordinate ring of a smooth projective curve over a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
and ''A'' the affine coordinate ring of a singular model. The ring ''A'' does not have unique factorization into prime ideals, and the failure of unique factorization is measured by the conductor
.
Ideals
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...
to the conductor share many of pleasant properties of ideals in Dedekind domains. Furthermore, for these ideals there is a tight correspondence between ideals of ''B'' and ideals of ''A'':
* The ideals of ''A'' that are relatively prime to
have unique factorization into products of invertible prime ideals that are coprime to the conductor. In particular, all such ideals are invertible.
* If ''I'' is an ideal of ''B'' that is relatively prime to
, then is an ideal of ''A'' that is relatively prime to
and the natural
ring homomorphism
In ring theory, a branch of abstract algebra, 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 such that ''f'' is:
:addition prese ...
is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. In particular, ''I'' is prime if and only if is prime.
* If ''J'' is an ideal of ''A'' that is relatively prime to
, then is an ideal of ''B'' that is relatively prime to
and the natural ring homomorphism
is an isomorphism. In particular, ''J'' is prime if and only if ''JB'' is prime.
* The functions
and
define a
bijection between ideals of ''A'' relatively prime to
and ideals of ''B'' relatively prime to
. This bijection preserves the property of being prime. It is also multiplicative, that is,
and
.
All of these properties fail in general for ideals not coprime to the conductor. To see some of the difficulties that may arise, assume that ''J'' is a non-zero ideal of both ''A'' and ''B'' (in particular, it is contained in, hence not coprime to, the conductor). Then ''J'' cannot be an invertible
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 dom ...
of ''A'' unless . Because ''B'' is a Dedekind domain, ''J'' is invertible in ''B'', and therefore
:
since we may multiply both sides of the equation by ''J''
−1. If ''J'' is also invertible in ''A'', then the same reasoning applies. But the left-hand side of the above equation makes no reference to ''A'' or ''B'', only to their shared fraction field, and therefore . Therefore being an ideal of both ''A'' and ''B'' implies non-invertibility in ''A''.
Conductors of quadratic number fields
Let ''K'' be a
quadratic extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
of Q, and let be its ring of integers. By extending to a Z-basis, we see that every order ''O'' in ''K'' has the form for some positive
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 ...
''c''. The conductor of this order equals the ideal ''cO''
''K''. Indeed, it is clear that ''cO''
''K'' is an ideal of ''O''
''K'' contained in ''O'', so it is contained in the conductor. On the other hand, the ideals of ''O'' containing ''cO''
''K'' are the same as ideals of the
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. I ...
. The latter ring is isomorphic to by the
second isomorphism theorem, so all such ideals of ''O'' are the sum of ''cO''
''K'' with an ideal of Z. Under this isomorphism, the conductor annihilates , so it must be .
In this case, the index is also equal to ''c'', so for orders of quadratic number fields, the index may be identified with the conductor. This identification fails for higher degree number fields.
Reference
{{Reflist
See also
*
Integral element 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'' ...
Ring theory
Commutative algebra