Smallest integer n for which n equals 0 in a ring
In mathematics, the characteristic of a ring "R", often denoted char("R"), is defined to be the smallest positive number of copies of the ring's multiplicative identity (1) that will sum to the additive identity (0). If no such number exists, the ring is said to have characteristic zero. 
That is, char("R") is the smallest positive number "n" such that:(p198, Thm. 23.14) 
 formula_1
if such a number "n" exists, and 0 otherwise.
Motivation.
The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, 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 "n" such that:(p198, Def. 23.12)
 formula_2
for every element "a" of the ring (again, if "n" exists; otherwise zero). This definition applies in the more general class of rngs (see ""); for (unital) rings the two definitions are equivalent due to their distributive law.
Equivalent characterizations.
 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".
 0 for all "a" ∈ "R" implies that "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 zero ring, which has only a single element 0. 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 formula_5 of integers modulo "n" has characteristic "n". If "R" is a subring of "S", then "R" and "S" have the same characteristic. For example, if "p" is prime and "q"("X") is an irreducible polynomial with coefficients in the field formula_7 with p elements, then the quotient ring formula_8 is a field of characteristic "p". Another example: The field formula_9 of complex numbers contains formula_3, so the characteristic of formula_9 is 0.
A formula_5-algebra is equivalently a ring whose characteristic divides "n". This is because for every ring "R" there is a ring homomorphism formula_13, and this map factors through formula_5 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 normally incorrect "freshman's dream" holds for power "p".
The map "x" ↦ "x""p" then defines a ring homomorphism "R" → "R", which 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". The "characteristic exponent" is defined similarly, except that it is equal to 1 when the characteristic is 0; otherwise it has the same value as the characteristic.
Any field "F" has a unique minimal subfield, also called its ;templatestyles src="Template:Visible anchor/styles.css" / ;prime field. This subfield is isomorphic to either the rational number field formula_15 or a finite field formula_7 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.
Fields of characteristic zero.
The most common fields of "characteristic zero" 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, such as the field of rational numbers formula_15 or the field of real numbers formula_18, the characteristic is 0. Thus, every algebraic number field and the field of complex numbers formula_9 are of characteristic zero.
Fields of prime characteristic.
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 formula_20, the algebraic closure of formula_20 or the field of formal Laurent series formula_22.
The size of any finite ring of prime characteristic "p" is a power of "p". Since in that case it contains formula_20 it is also 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.
References.
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