Characteristic (algebra)
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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 (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity 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 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 Multiplicative identity: mandatory vs. optional), 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 the unique ring homomorphism from Z to R;^{[1]}
- The characteristic is the natural number n such that R contains a subring isomorphic to the factor ring Z/nZ, which is the image of the above 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 precisely if the statement ka = 0 for all a ∈ R implies k is a multiple of n.
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 an irreducible 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 integers, their characteristic is 0.
A Z/nZ-algebra is equivalently a ring whose characteristic divides n. This is because for every ring R there is a ring homomorphism Z → R, and this map factors through Z/nZ if and only if the characteristic of R divides n. In this case 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 positive characteristic or prime 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 (ring-)isomorphic to a subfield of complex numbers).^{[2]} 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(p^{n}) 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.^{[3]}
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}.)
References
- ^ 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).
- ^ 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.
- ^ "Field Characteristic Exponent". Wolfram Mathworld. Wolfram Research. Retrieved May 27, 2015.
- Neal H. McCoy (1964, 1973) The Theory of Rings, Chelsea Publishing, page 4.