Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest positive number of copies of the ring's multiplicative identity (1) that will sum to the additive identity (0). If no such number exists, the ring is said to have characteristic zero.

That is, char(R) is the smallest positive number n such that:

if such a number n exists, and 0 otherwise.

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:

for every element a of the ring (again, if n exists; otherwise zero). This definition applies in the more general class of rngs (see Ring (mathematics) § Multiplicative identity and the term "ring"); for (unital) rings the two definitions are equivalent due to their distributive law.

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 zero ring, which has only a single element 0. 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 zero is infinite.

The ring ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ of integers modulo n has characteristic n. If R is a subring of S, then R and S have the same characteristic. For example, if p is prime and q(X) is an irreducible polynomial with coefficients in the field ${\displaystyle \mathbb {F} _{p}}$ with p elements, then the quotient ring ${\displaystyle \mathbb {F} _{p}[X]/(q(X))}$ is a field of characteristic p. Another example: The field ${\displaystyle \mathbb {C} }$ of complex numbers contains ${\displaystyle \mathbb {Z} }$, so the characteristic of ${\displaystyle \mathbb {C} }$ is 0.