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

1
+
⋯
+
1
⏟
n
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

a
+
⋯
+
a
⏟
n
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.

The characteristic is the natural number *n* such that *n***Z** 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**/*n***Z**, 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).

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**/*n***Z** 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**/*p***Z** where *p* is prime, then the factor ring (**Z**/*p***Z**)[*X*] / (*q*(*X*)) is a field of characteristic *p*. Since the complex numbers contain the rationals, their characteristic is 0.

A **Z**/*n***Z**-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.

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 isomorphic to a subfield of complex numbers). 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**/*p***Z**, the algebraic closure of **Z**/*p***Z** or the field of formal Laurent series **Z**/*p***Z**((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.

The size of any finite ring of prime characteristic *p* is a power of *p*. Since in that case it must contain **Z**/*p***Z** 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}.)