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:[1](p 198, Thm. 23.14)

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, that is, the smallest positive integer n such that:[1](p 198, Def. 23.12)

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 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.

If R and S are rings and there exists a ring homomorphism RS, 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 zero 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.

If a commutative ring R has prime characteristic p, then we have (x + y)p = xp + yp for all elements x and y in R – the normally incorrect "freshman's dream" holds for power p. The map f (x) = xp then defines a ring homomorphism RR . It is called the Frobenius homomorphism. If R is an integral domain it is injective.

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.