Characteristic (algebra) - Case of Rings

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 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. Any ring of characteristic 0 is infinite.

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)/(q(X)) is a field of characteristic p. Since the complex numbers contain the rationals, their characteristic is 0.

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 "freshman's dream" holds for power p.

The map

f(x) = xp

then defines a ring homomorphism

R → R.

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