In
mathematics, the characteristic of a
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 ...
, often denoted , is defined to be the smallest number of times one must use the ring's
multiplicative identity (1) in a sum to get the
additive identity In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from elem ...
(0). If this sum never reaches the additive identity the ring is said to have characteristic zero.
That is, is the smallest positive number such that:
[
:
if such a number exists, and otherwise.
]
Motivation
The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately.
The characteristic may also be taken to be the exponent of the ring's 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 structure ...
, that is, the smallest positive integer such that:[
]
:
for every element of the ring (again, if exists; otherwise zero). Some authors do not include the multiplicative identity element in their requirements for a ring (see Multiplicative identity and the term "ring"), and this definition is suitable for that convention; otherwise the two definitions are equivalent due to the distributive law in rings.
Equivalent characterizations
* The characteristic is the natural number such that is the kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learni ...
of the unique ring homomorphism from to .
* The characteristic is the natural number such that contains a subring isomorphic to the factor ring , which is the image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of the above homomorphism.
* When the non-negative integers are partially ordered by divisibility, then is the smallest and is the largest. Then the characteristic of a ring is the smallest value of for which If nothing "smaller" (in this ordering) than will suffice, then the characteristic is . This is the appropriate partial ordering because of such facts as that is the least common multiple of and , and that no ring homomorphism exists unless divides
* The characteristic of a ring is precisely if the statement for all implies is a multiple of .
Case of rings
If ''R'' and ''S'' are rings and there exists a ring homomorphism , then the characteristic of divides the characteristic of . This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic 1 is the zero ring, which has only a single element If a nontrivial ring does not have any nontrivial zero divisors, then its characteristic is either or prime
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 ways ...
. In particular, this applies to all fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
*Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...
, to all 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 se ...
s, and to all division rings. Any ring of characteristic is infinite.
The ring of integers modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is ...
has characteristic . If is a subring of , then and have the same characteristic. For example, if is prime and is an irreducible polynomial with coefficients in the field with elements, then the quotient ring is a field of characteristic . Another example: The field of complex numbers contains , so the characteristic of is .
A -algebra is equivalently a ring whose characteristic divides . This is because for every ring there is a ring homomorphism , and this map factors through if and only if the characteristic of divides . In this case for any in the ring, then adding to itself times gives .
If a commutative ring has ''prime characteristic'' , then we have for all elements and in – the normally incorrect " freshman's dream" holds for power .
The map then defines a ring homomorphism It is called the ''Frobenius homomorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism ma ...
''. If 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 se ...
it is injective.
Case of fields
As mentioned above, the characteristic of any field is either or a prime number. A field of non-zero characteristic is called a field of ''finite characteristic'' or ''positive characteristic'' or ''prime characteristic''. The ''characteristic exponent'' is defined similarly, except that it is equal to if the characteristic is ; otherwise it has the same value as the characteristic.[
]
Any field has a unique minimal subfield, also called its . This subfield is isomorphic to either the rational number field or a finite field of prime order. Two prime fields of the same characteristic are isomorphic, and this isomorphism is unique. In other words, there is essentially a unique prime field in each characteristic.
Fields of characteristic zero
The most common fields of ''characteristic zero'' that are the subfields of the complex numbers. The p-adic field
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extens ...
s are characteristic zero fields that are widely used in number theory. They have absolute values which are very different from those of complex numbers.
For any ordered field, as the field of rational numbers or the field of real numbers , the characteristic is . Thus, every 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 f ...
and the field of complex numbers are of characteristic zero.
Fields of prime characteristic
The finite field GF() has characteristic ''p''.
There exist infinite fields of prime characteristic. For example, the field of all rational functions over , the algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
of or the field of formal Laurent series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...
.
The size of any finite ring
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements.
Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite gro ...
of prime characteristic is a power of . Since in that case it contains it is also a vector space over that field, and from linear algebra we know that the sizes of finite vector spaces over finite fields are a power of the size of the field. This also shows that the size of any finite vector space is a prime power.
Notes
References
Sources
*
{{refend
Ring theory
Field (mathematics)