# Characteristic

In mathematics, the**characteristic**of a ring

*R*with identity element 1

_{R}is defined to be the smallest positive integer

*n*such that

*n*1

_{R}= 0 (where

*n*1

_{R}is defined as 1

_{R}+ ... + 1

_{R}with

*n*summands). If no such

*n*exists, we say that the characteristic of

*R*is 0. The characteristic of

*R*is often denoted char(

*R*).

The characteristic of the ring *R* may be equivalently defined as the unique natural number *n* such that *n***Z** is the kernel of the unique ring homomorphism from **Z** to *R* which sends 1 to 1_{R}. And yet another equivalent definition: the characteristic of *R* is the unique natural number *n* such that *R* contains a subring isomorphic to the factor ring **Z**/*n***Z**.

## Examples and notes

- 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 the non-trivial ring
*R*does not have any 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. - For any ordered field (for example, the rationals or the reals) the characteristic is 0.
- 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 (**Z**/*p***Z**)[*X*]/(*q*(*X*)) is a field of characteristic*p*. Since the complex numbers contain the rationals, their characteristic is 0. - Any ring of characteristic 0 is infinite. 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**is one such. The algebraic closure of**Z**/*p***Z**is another example. - 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}.) - 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 map*f*(*x*) =*x*^{p}defines an injective ring homomorphism*R*`->`*R*. It is called the*Frobenius homomorphism*.

The term "characteristic" is also used in several other unrelated mathematical contexts:

**Characteristic**is also sometimes used as a piece of jargon in discussions of universals in metaphysics, often in the phrase 'distinguishing characteristics'.