, the characteristic
of a ring R
with identity element
is defined to be the smallest positive integer n
such that n
= 0 (where n
is defined as 1R
+ ... + 1R
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 nZ is the kernel of the unique ring homomorphism from Z to R which sends 1 to 1R. 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/nZ.
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/nZ 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/pZ where p is prime, then the factor ring (Z/pZ)[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(pn) has characteristic p.
- There exist infinite fields of prime characteristic. For example, the field of all rational functions over Z/pZ is one such. The algebraic closure of Z/pZ 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/pZ 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 pn. So its size is (pn)m = pnm.)
- 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 map f(x) = xp 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'.