Characteristic

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[edit]

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

0  | = 0 | = | 0+0+0+0+0+0

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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.