Last modified on July 13, 2016, at 05:53

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

  • - 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.