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 element (1) in a sum to get the additive identity element (0); the ring is said to have characteristic zero if this repeated sum never reaches the additive identity.
That is, char(R) is the smallest positive number n such that
if such a number n exists, and 0 otherwise.
The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive n such that
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 ring), and this definition is suitable for that convention; otherwise the two definitions are easily seen to be equivalent due to the distributive law in rings.
Other equivalent definitions include taking the characteristic to be the natural number n such that nZ is the kernel of a ring homomorphism from Z to R, or such that R contains a subring isomorphic to the factor ring Z/nZ, which would be the image of that homomorphism. The requirements of ring homomorphisms are such that there can be only one homomorphism from the ring of integers to any ring; in the language of category theory, Z is an initial object of the category of rings. Again this follows the convention that a ring has a multiplicative identity element (which is preserved by ring homomorphisms).
Contents
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 the nontrivial 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)[X]/(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} = x^{p} + y^{p} for all elements x and y in R – the "freshman's dream" holds for power p.
The map
 f(x) = x^{p}
then defines a ring homomorphism
 R → R.
It is called the Frobenius homomorphism. If R is an integral domain it is injective.
Case of fields
As mentioned above, the characteristic of any field is either 0 or a prime number.
For any field F, there is a minimal subfield, namely the prime field, the smallest subfield containing 1_{F}. It is isomorphic either to the rational number field Q, or a finite field of prime order, F_{p}; the structure of the prime field and the characteristic each determine the other. Fields of characteristic zero have the most familiar properties; for practical purposes they resemble subfields of the complex numbers (unless they have very large cardinality, that is; in fact, any field of characteristic zero and cardinality at most continuum is isomorphic to a subfield of complex numbers). The padic fields are characteristic zero fields, much applied in number theory, that are constructed from rings of characteristic p^{k}, as k → ∞.
For any ordered field (for example, the rationals or the reals) the characteristic is 0. 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/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 p^{n}. So its size is (p^{n})^{m} = p^{nm}.)
See also
 Characteristic exponent of a field
References
 Neal H. McCoy (1964, 1973) The Theory of Rings, Chelsea Publishing, page 4.
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