In mathematics, the characteristic of a ring , often denoted , 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,  is the smallest positive number  such that:
\underbrace{1+\cdots+1}_{n \text{ summands}} = 0
if such a number  exists, and  otherwise.
Motivation
The special definition of the characteristic zero is motivated by the equivalent definitions given in , where the characteristic zero is not required to be considered separately.
The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer  such that:
\underbrace{a+\cdots+a}_{n \text{ summands}} = 0
for every element  of the ring (again, if  exists; otherwise zero).
Some authors do not include the multiplicative identity element in their requirements for a ring (see Multiplicative identity and the term "ring"), and this definition is suitable for that convention; otherwise the two definitions are equivalent due to the distributive law in rings.
Equivalent characterizations
The characteristic is the natural number  such that  is the kernel of the unique ring homomorphism from  to ;
The characteristic is the natural number  such that  contains a subring isomorphic to the factor ring , which is the image of the above homomorphism.
When the non-negative integers  are partially ordered by divisibility, then  is the smallest and  is the largest.
Then the characteristic of a ring is the smallest value of  for which  If nothing "smaller" (in this ordering) than  will suffice, then the characteristic is .
This is the appropriate partial ordering because of such facts as that  is the least common multiple of  and  and that no ring homomorphism  exists unless  divides
The characteristic of a ring  is  precisely if the statement  for all  implies  is a multiple of .
Case of rings
If R and S are rings and there exists a ring homomorphism , then the characteristic of  divides the characteristic of .
This can sometimes be used to exclude the possibility of certain ring homomorphisms.
The only ring with characteristic 1 is the zero ring, which has only a single element  If a nontrivial ring  does not have any nontrivial zero divisors, then its characteristic is either  or prime.
In particular, this applies to all fields, to all integral domains, and to all division rings.
Any ring of characteristic  is infinite.
The ring  of integers modulo  has characteristic .
If  is a subring of , then  and  have the same characteristic.
For example, if  is prime and  is an irreducible polynomial with coefficients in the field  then the quotient ring  is a field of characteristic .
Another example: The field  of complex numbers contains , so the characteristic of  is .
A -algebra is equivalently a ring whose characteristic divides .
This is because for every ring  there is a ring homomorphism , and this map factors through  if and only if the characteristic of  divides .
In this case for any  in the ring, then adding  to itself  times gives
If a commutative ring  has prime characteristic , then we have  for all elements  and  in  – the normally incorrect "freshman's dream" holds for power .
The map  then defines a ring homomorphism  It is called the Frobenius homomorphism.
If  is an integral domain it is injective.
Case of fields {{anchor|Fields}}
As mentioned above, the characteristic of any field is either  or a prime number.
A field of non-zero characteristic is called a field of finite characteristic or positive characteristic or prime characteristic.
Any field  has a unique minimal subfield, also called its .
This subfield is isomorphic to either the rational number field  or a finite field  of prime order.
Two prime fields of the same characteristic are isomorphic, and this isomorphism is unique.
In other words, there is essentially a unique prime field in each characteristic.
The most common fields of characteristic zero that are the subfields of the complex numbers.
The p-adic fields are characteristic zero fields that are widely used in number theory.
They have absolute values which are very different from those of complex numbers.
For any ordered field, as the field of rational numbers  or the field of real numbers , the characteristic is .
Thus, number fields and the field of complex numbers  are of characteristic zero.
Actually, every field of characteristic zero is the quotient field of a ring  where  is a set of variables and  a set of polynomials in .
The finite field GF() has characteristic p.
There exist infinite fields of prime characteristic.
For example, the field of all rational functions over , the algebraic closure of  or the field of formal Laurent series .
The characteristic exponent is defined similarly, except that it is equal to  if the characteristic is ; otherwise it has the same value as the characteristic.
The size of any finite ring of prime characteristic  is a power of .
Since in that case it must contain  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.
Notes
References
Sources
