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 sum never reaches the additive identity.

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

$\underbrace{1+\cdots+1}_{n \text{ summands}} = 0$

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

$\underbrace{a+\cdots+a}_{n \text{ summands}} = 0$

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 equivalent due to the distributive law in rings.

Other equivalent characterizations

• The characteristic is the natural number n such that nZ is the kernel of a ring homomorphism from Z to R;
• The characteristic is the natural number n such that R contains a subring isomorphic to the factor ring Z/nZ, which would be the image of that homomorphism.
• When the non-negative integers {0, 1, 2, 3, ...} are partially ordered by divisibility, then 1 is the smallest and 0 is the largest. Then the characteristic of a ring is the smallest value of n for which n ⋅ 1 = 0. If nothing "smaller" (in this ordering) than 0 will suffice, then the characteristic is 0. This is the appropriate partial ordering because of such facts as that char(A × B) is the least common multiple of char A and char B, and that no ring homomorphism f : A → B exists unless char B divides char A.
• The characteristic of a ring R is n ∈ {0, 1, 2, 3, ...} precisely if the statement ka = 0 for all a ∈ R implies n is a divisor of k.

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).

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 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.

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.

A Z/nZ-algebra is equivalently a ring whose characteristic divides n. For any r in the ring, then adding r to itself n times gives nr = 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. A field of non-zero characteristic is called a field of finite characteristic or a field of positive characteristic.

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 to 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).^[1] The p-adic fields or any finite extension of them are characteristic zero fields, much applied in number theory, that are constructed from rings of characteristic p^k, as k → ∞.

For any ordered field, as the field of rational numbers Q or the field of real numbers R, the characteristic is 0. Thus, number fields and the field of complex numbers C are of characteristic zero. Actually, every field of characteristic zero is the quotient field of a ring Q[X]/P where X is a set of variables and P a set of polynomials in Q[X]. 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, the algebraic closure of Z/pZ or the field of formal Laurent series Z/pZ((T)). The characteristic exponent is defined similarly, except that it is equal to 1 if the characteristic is zero; otherwise it has the same value as the characteristic.^[2]

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 = p^nm.)

References

1. Enderton, Herbert B. (2001), A Mathematical Introduction to Logic (2nd ed.), Academic Press, p. 158, ISBN 9780080496467. Enderton states this result explicitly only for algebraically closed fields, but also describes a decomposition of any field as an algebraic extension of a transcendental extension of its prime field, from which the result follows immediately.
2. "Field Characteristic Exponent". Wolfram Mathworld. Wolfram Research. Retrieved May 27, 2015.

• Neal H. McCoy (1964, 1973) The Theory of Rings, Chelsea Publishing, page 4.