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: if such a number exists, and 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 such that: 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.