Characteristic (algebra)

Characteristic (algebra)

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 repeated sum never reaches the additive identity.

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 ring), and this definition is suitable for that convention; otherwise the two definitions are easily seen to be equivalent due to the distributive law in rings.