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*characteristic (algebra)*

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characteristic (algebra)

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**characteristic**of a ring

*R*, often denoted char(

*R*), 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 (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, char(

*R*) is the smallest positive number

*n*such that

underbrace{1+cdots+1}_{n text{ summands}} = 0

if such a number *n*exists, and 0 otherwise.The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive

*n*such that

underbrace{a+cdots+a}_{n text{ summands}} = 0

for every element *a*of the ring (again, if

*n*exists; otherwise zero). Some authors do not include the multiplicative identity element in their requirements for a ring (see (ring (mathematics)#Multiplicative identity: mandatory vs. optional|Multiplicative identity: mandatory vs. optional)), and this definition is suitable for that convention; otherwise the two definitions are equivalent due to the distributive law in rings.

## Other equivalent characterizations

- The characteristic is the natural number
*n*such that*n***Z**is the kernel of the unique ring homomorphism from**Z**to*R*;The requirements of ring homomorphisms are such that there can be only one homomorphism from the ring of integers to any ring; in the language of category theory,**Z**is an initial object of the category of rings. Again this follows the convention that a ring has a multiplicative identity element (which is preserved by ring homomorphisms). - The characteristic is the natural number
*n*such that*R*contains a subring isomorphic to the factor ring**Z**/*n***Z**, which is the image of the above homomorphism. - When the non-negative integers {{nowrap|{0, 1, 2, 3, ...}{{null}}}} are partially ordered by divisibility, then 1 is the smallest and 0 is the largest. Then the characteristic of a ring is the smallest value of
*n*for which {{nowrap|1=*n*⋅ 1 = 0}}. If nothing "smaller" (in this ordering) than 0 will suffice, then the characteristic is 0. This is the appropriate partial ordering because of such facts as that {{nowrap|char(*A*Ã—*B*)}} is the least common multiple of {{nowrap|char*A*}} and {{nowrap|char*B*}}, and that no ring homomorphism {{nowrap|*f*:*A*â†’*B*}} exists unless {{nowrap|char*B*}} divides {{nowrap|char*A*}}. - The characteristic of a ring
*R*is {{nowrap|*n*âˆˆ {0, 1, 2, 3, ...}{{null}}}} precisely if the statement {{nowrap|1=*ka*= 0}} for all {{nowrap|*a*âˆˆ*R*}} implies*n*is a divisor of*k*.

## Case of rings

If*R*and

*S*are rings and there exists a ring homomorphism

*R*â†’

*S*, then the characteristic of

*S*divides the characteristic of

*R*. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic 1 is the trivial ring, which has only a single element {{nowrap|1=0 = 1}}. If a nontrivial ring

*R*does not have any nontrivial zero divisors, then its characteristic is either 0 or prime. In particular, this applies to all fields, to all integral domains, and to all division rings. Any ring of characteristic 0 is infinite.The ring

**Z**/

*n*

*Z***of integers modulo****n***has characteristic*n*. If*R*is a subring of*S*, then*R*and*S*have the same characteristic. For instance, if*q*(*X*) is a prime polynomial with coefficients in the field**Z***/****p****Z**where

*p*is prime, then the factor ring {{nowrap|(

**Z**/

*p*

**Z**)[

*X*] / (

*q*(

*X*))}} is a field of characteristic

*p*. Since the complex numbers contain the rationals, their characteristic is 0.A

**Z**/

*n*

*Z***-algebra is equivalently a ring whose characteristic divides****n***. This is because for every ring*R*there is a ring homomorphism**Z***â†’****R***, and this map factors through**Z***/****n****Z**if and only if the characteristic of

*R*divides

*n*. In this case for any

*r*in the ring, then adding

*r*to itself

*n*times gives {{nowrap|1=

*nr*=

*0*}}.If a commutative ring

*R*has

**prime characteristic**

*p*, then we have {{nowrap|1=(

*x*+

*y*)

*p*=

*x*

**'p****+**

*y**'p*}} for all elements

*x*and

*y*in

*R*â€“ the "freshman's dream" holds for power

*p*.The map

*f*(

*x*) =

*x*

*p*

*R*â†’

*R*.

*Frobenius homomorphism*. If

*R*is an integral domain it is injective.

## Case of fields {{anchor|Fields}}

As mentioned above, the characteristic of any field is either 0 or a prime number. A field of non-zero characteristic is called a field of*finite characteristic*or

*positive characteristic*or

*prime characteristic*.For any field

*F*, there is a minimal subfield, namely the

**{{visible anchor|prime field}}**, the smallest subfield containing 1

*F*. It is isomorphic either to the rational number field

**Q**, or to a finite field of prime order,

**F**

*p*; the structure of the prime field and the characteristic each determine the other. Fields of

*characteristic zero*have the most familiar properties; for practical purposes they resemble subfields of the complex numbers (unless they have very large cardinality, that is; in fact, any field of characteristic zero and cardinality at most continuum is (ring-)isomorphic to a subfield of complex numbers).{{citation|title=A Mathematical Introduction to Logic|first=Herbert B.|last=Enderton|authorlink=Herbert Enderton|edition=2nd|publisher=Academic Press|year=2001|isbn=9780080496467|url=https://books.google.com/books?id=dVncCl_EtUkC&pg=PA158|page=158}}. Enderton states this result explicitly only for algebraically closed fields, but also describes a decomposition of any field as an algebraic extension of a transcendental extension of its prime field, from which the result follows immediately. The p-adic fields or any finite extension of them are characteristic zero fields, much applied in number theory, that are constructed from rings of characteristic

*p*

*k*, as

*k*â†’ âˆž.For any ordered field, as the field of rational numbers

**Q**or the field of real numbers

**R**, the characteristic is 0. Thus, number fields and the field of complex numbers

**C**are of characteristic zero. Actually, every field of characteristic zero is the quotient field of a ring

**Q**[X]/P where X is a set of variables and P a set of polynomials in

**Q**[X]. The finite field GF(

*p*

**'n****) has characteristic**Z

*p*. There exist infinite fields of prime characteristic. For example, the field of all rational functions over**/**Z

*p**Z**, the algebraic closure of*p**Z**/**or the field of formal Laurent series**Z

**/**Z

*p**'((T)). The*characteristic exponent'' is defined similarly, except that it is equal to 1 if the characteristic is zero; otherwise it has the same value as the characteristic.WEB,weblink Field Characteristic Exponent, Wolfram Mathworld, Wolfram Research, May 27, 2015, The size of any finite ring of prime characteristic

*p*is a power of

*p*. Since in that case it must contain

**Z**/

*p*

**Z**it must also be 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. (It is a vector space over a finite field, which we have shown to be of size

*p*

**'n****, so its size is (**

*p**'n*)

*m*=

*p*

*nm*.)

## References

{{Reflist}}- Neal H. McCoy (1964, 1973)
*The Theory of Rings*, Chelsea Publishing, page 4.

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