In algebra, the **characteristic** of a ring *R* is the least natural number *n* such that, for all *r* in *R*, *n* * *r* = 0. If no such natural number exists, the ring has characteristic 0. The characteristic of a field must be either 0 or a prime number.

## Examples

- - the set of the integers - has characteristic 0.
- has characteristic 6:

0 = 0 = 0+0+0+0+0+0 1+1+1+1+1+1 = 0 = 1+1+1+1+1+1 2+2+2 = 0 = 2+2+2+2+2+2 3+3 = 0 = 3+3+3+3+3+3 4+4+4 = 0 = 4+4+4+4+4+4 5+5+5+5+5+5 = 0 = 5+5+5+5+5+5

- If
*R*is a ring with unity, i.e., there exists a multiplicative identity**1**, then the characteristic is already given by the smallest*r*such that*r****1**= 0. If no such*r*exists, the characteristic is defined as zero.