# Characteristic (algebra)

In mathematics, the **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

- [math]\displaystyle{ \underbrace{1+\cdots+1}_{n \text{ summands}} = 0 }[/math]

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

- [math]\displaystyle{ \underbrace{a+\cdots+a}_{n \text{ summands}} = 0 }[/math]

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 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*;^{[1]} - 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 {0, 1, 2, 3, ...} 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*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 char(*A*×*B*) is the least common multiple of char*A*and char*B*, and that no ring homomorphism*f*:*A*→*B*exists unless char*B*divides char*A*. - The characteristic of a ring
*R*is*n*precisely if the statement*ka*= 0 for all*a*∈*R*implies*k*is a multiple of*n*.

## Case of rings

If *R* and *S* are rings and there exists a ring homomorphism *R* → *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 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 an irreducible polynomial with coefficients in the field **Z**/*p***Z** where *p* is prime, then the factor ring (**Z**/*p***Z**)[*X*] / (*q*(*X*)) is a field of characteristic *p*. Since the complex numbers contain the integers, 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** → *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 *nr* = *0*.

If a commutative ring *R* has **prime characteristic** *p*, then we have (*x* + *y*)^{p} = *x*^{p} + *y*^{p} for all elements *x* and *y* in *R* – the "freshman's dream" holds for power *p*.

The map

*f*(*x*) =*x*^{p}

then defines a ring homomorphism

*R*→*R*.

It is called the *Frobenius homomorphism*. If *R* is an integral domain it is injective.

## Case of 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 **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).^{[2]} 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* → ∞.

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 *p*. There exist infinite fields of prime characteristic. For example, the field of all rational functions over **Z**/*p***Z**, the algebraic closure of **Z**/*p***Z** or the field of formal Laurent series **Z**/*p***Z**((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.^{[3]}

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

- ↑ 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). - ↑ Enderton, Herbert B. (2001),
*A Mathematical Introduction to Logic*(2nd ed.), Academic Press, p. 158, ISBN 9780080496467, https://books.google.com/books?id=dVncCl_EtUkC&pg=PA158. 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. - ↑ "Field Characteristic Exponent". Wolfram Research. http://mathworld.wolfram.com/FieldCharacteristicExponent.html.

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

Original source: https://en.wikipedia.org/wiki/Characteristic (algebra).
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